Conditional measure on the Brownian path and other random sets

TL;DR

This paper investigates the existence and construction of a random measure supported on Brownian motion paths in high dimensions, satisfying a prescribed expectation condition, and extends the theory to more general random sets.

Contribution

It provides conditions for the existence of such measures, explicit constructions when possible, and generalizes the framework to broader classes of random sets and applications.

Findings

Explicit formula for the measure when u is Lebesgue measure

Calculation of the expectation of the double integral w.r.t. the measure

Extension of the theory to general random sets in metric spaces

Abstract

Let $B$ denote the range of the Brownian motion in $\mathbb{R}^{d}$ ($d\geq3$). For a deterministic Borel measure $\nu$ on $\mathbb{R}^{d}$ we wish to find a random measure $\mu$ such that the support of $\mu$ is contained in $B$ and it is a solution to the equation $E(\mu(A))=\nu(A)$ for every Borel set $A$. We discuss when it is possible to find a solution $\mu$ and in that case we construct the solution. We study several properties of $\mu$ such as the probability of $\mu\neq0$ and we establish a formula for the expectation of the double integral with respect to $\mu\times\mu$. We calculate $\mu$ in terms of the occupation measure when $\nu$ is the Lebesgue measure, i.e. we provide an explicit deterministic density function of $\mu$ with respect to the occupation measure. As a conclusion we calculate an explicit formula for the expectation of the double integral with respect to the occupation measure. We generalise the theory for more general random sets in separable, metric, Radon spaces. As an additional example, we also apply our results to percolation limit sets on boundaries of trees.

Equations

$$\frac{\nu {|}_{{\pi }^{-1}B(x,r)}}{2r}$$

$$\mathrm{d}\nu (x,y)=\mathrm{d}{\mu }_x(x,y)\mathrm{d}\lambda (x),$$

$${\nu }_x={\left(\frac{\mathrm{d}{\pi }_{\ast }\nu }{\mathrm{d}\lambda }(x)\right)}^{-1}\cdot {\mu }_x$$

$$\int f\mathrm{d}{\mu }_x(z)=\underset{r\to 0}{\lim }(2r{)}^{-1}\underset{{\pi }^{-1}(B(x,r))}{\int }f\mathrm{d}\nu (z)$$

$$\mathrm{d}\nu (z)=\mathrm{d}\mu (z)\mathrm{d}P(x)$$

$${\mu }_k=\frac{\nu {|}_{{\pi }^{-1}\left(\pi \left(Q_k(x,0)\right)\right)}}{2^{-k}}=\sum _{Q\in {\mathcal{Q}}_k}P(Q\cap B\ne \varnothing {)}^{-1}\cdot I_{Q\cap B\ne \varnothing }\cdot \nu {|}_Q$$

$$\underset{k\to \infty }{\lim }{\mu }_k(A)=\mu (A)$$

$$\underset{k\to \infty }{\lim }{\mu }_k([0,1{]}^2)=0$$

$$\nu ={\nu }_R+{\nu }_{\perp }$$

$$\mathrm{d}{\nu }_R(z)=\mathrm{d}\mu (z)\mathrm{d}P.$$

$$\mathrm{d}\mu (z)\mathrm{d}P=\mathrm{d}{\nu }_R(z)$$

$$\mathrm{d}\mu (x)\mathrm{d}\mu (y)\mathrm{d}P=\frac{{\Vert x\Vert }^{d-2}+{\Vert y\Vert }^{d-2}}{{\Vert x-y\Vert }^{d-2}}\mathrm{d}{\nu }_R(x)\mathrm{d}{\nu }_R(y).$$

$$\int \underset{A}{\int }\mathrm{d}\mu (z)\mathrm{d}P=E(\mu (A))=0.$$

$$\nu (A)=\sup \left\{\nu (K):K\subseteq A\mathrm{compact}\right\}$$

$$\nu (A)=\inf \left\{\nu (U):A\subseteq U\mathrm{open}\right\}.$$

$$\underset{k\to \infty }{\lim }\sup \left\{\mathrm{diam}(Q):Q\in {\mathcal{Q}}_k\right\}=0,$$

$$Q\subseteq D.$$

$$\nu (X\setminus X_0)=0.$$

$$P(Q\cap B\ne \varnothing )>0$$

$${\mathcal{C}}_k(\nu )=\sum _{Q\in {\mathcal{Q}}_k}P(Q\cap B\ne \varnothing {)}^{-1}\cdot I_{Q\cap B\ne \varnothing }\cdot \nu {|}_Q.$$

$$E({\mathcal{C}}_k(\nu )(A))=\nu (A)$$

$$\phi (r)\le c_2\phi (r\cdot (1+2\delta ))+c_3$$

$$\phi (0)=\infty ,$$

$$\#\left\{S\in {\mathcal{Q}}_k:\max \left\{\mathrm{diam}(Q),\mathrm{diam}(S)\right\}\ge \delta \cdot \mathrm{dist}(Q,S)\right\}\le M_{\delta },$$

$$0<\mathrm{diam}(Q)/M\le \mathrm{diam}(S)\le \mathrm{diam}(Q)\cdot M<\infty$$

$$I_{\phi }(\nu )=\int \int \phi (x,y)\mathrm{d}\nu (x)\mathrm{d}\nu (y).$$

$$C_{\phi }(K)=\sup \left\{I_{\phi }(\nu {)}^{-1}:\nu \mathrm{is}\mathrm{a}\mathrm{Borel}\mathrm{probability}\mathrm{measure}\mathrm{on}\mathrm{K}\right\}.$$

$$a{C}_{\phi }(Q)\le P(Q\cap B\ne \varnothing )$$

$$P(Q\cap B\ne \varnothing \mathrm{and}S\cap B\ne \varnothing )\le c\cdot P(Q\cap B\ne \varnothing )\cdot P(S\cap B\ne \varnothing )\cdot \phi (\mathrm{dist}(Q,S)).$$

$$f(x)\mathrm{d}\nu (x)$$

Full text

1 Introduction

We begin by the analogy of slicing measures with straight lines to introduce the concept. Let $\nu$ be a finite Borel probability measure on the unit square of the plane, let $\pi(x,y)=x$ be the projection onto the $x$-axis and $\pi_{*}\nu=\nu\circ\pi^{-1}$ be the projection measure. Then by the existence of the regular conditional measure [8, Theorem 5.1.9] for $\pi_{*}\nu$ almost every $x\in[0,1]$ there exists a Borel probability measure $\nu_{x}$ on the slice $\{x\}\times[0,1]$ such that we get the disintegration formula $\mathrm{d}\nu(x,y)=\mathrm{d}\nu_{x}(x,y)\mathrm{d}\pi_{*}\nu(x)$. When $\pi_{*}\nu\ll\lambda$, where $\lambda$ denotes the $1$-dimensional Lebesgue measure, the measure $\nu_{x}$ can be obtained as a weak limit of certain rescaled restrictions of $\nu$. Assume that $\pi_{*}\nu\ll\lambda$, then for Lebesgue almost every $x\in[0,1]$ the weak limit of the measures

[TABLE]

exists as $r$ approaches [math], see [17, Chapter 10], where $\nu|_{A}$ denotes the restriction of $\nu$ to $A$, that is $\nu|_{A}(B)=\nu(A\cap B)$ for every Borel sets $A,B\subseteq\mathbb{R}^{2}$. Let this weak limit be $\mu_{x}$ for Lebesgue almost every $x\in[0,1]$, then

[TABLE]

see Mattila [18, Lemma 3.4]. Thus by the uniqueness of the conditional measure

[TABLE]

for $\pi_{*}\nu$ almost every $x\in[0,1]$, where $\frac{\mathrm{d}\nu}{\mathrm{d}\mu}(x)$ denotes the Radon-Nykodym derivative.

It was shown by Mattila [18, Lemma 3.4] that for a Borel function $f:[0,1]^{2}\longrightarrow\mathbb{R}$ with $\int\left|f\right|\mathrm{d}\nu(z)<\infty$ we have that

[TABLE]

for Lebesgue almost every $x\in[0,1]$. Mattila [17, Theorem 10.7] also discusses the double integral of certain kernels with respect to $\mu_{x}\times\mu_{x}$ which gives a strong tool to analyse the geometric measure theory of slices of measures and sets.

One can look at it as: We randomly choose a slice $B=\{x\}\times[0,1]$ where we choose $x$ uniformly in $[0,1]$. Then $\mu=\mu_{x}$ is a random measure supported on the random slice, and

[TABLE]

holds by (1.2), i.e. $\nu$ is the expectation of $\mu$. Our main goal in this paper is to construct this kind of slice measures on random slices but instead of taking the random slices to be straight line segments we take the slices to be the Brownian path or other random sets.

Let $Q_{k}(z)$ be the dyadic cube $[\frac{i_{1}}{2^{k}},\frac{i_{1}+1}{2^{k}})\times[\frac{i_{2}}{2^{k}},\frac{i_{2}+1}{2^{k}})$ for $i_{1},i_{2}\in\mathbb{Z}$ such that $z\in Q_{k}(z)$ for some $z\in[0,1]^{2}$ and let $\mathcal{Q}_{k}=\left\{Q_{k}(z):z\in[0,1]^{2}\right\}$. It can be shown that for Lebesgue almost every $x\in[0,1]$ we get the same weak limit $\mu_{x}$ if we consider the sequence

[TABLE]

instead of (1.1), where $I_{Q\cap B\neq\emptyset}$ is the indicator function of the event $Q\cap B\neq\emptyset$. We can obtain from the analogue of (1.3) that for every Borel set $A\subseteq\mathbb{R}^{2}$

[TABLE]

almost surely, i.e. for Lebesgue almost every slice.

If $\pi_{*}\nu$ is singular to the Lebesgue measure then by Lebesgue‘s density theorem

[TABLE]

for Lebesgue almost every $x\in[0,1]$, i.e. $\mu=0$ almost surely. Hence in general we can decompose $\nu$ into two parts

[TABLE]

such that $\pi_{*}\nu_{R}\ll\lambda$ and $\pi_{*}\nu_{\perp}\perp\lambda$, one part corresponds to a vanishing limit (1.7), the other part corresponds to an $\mathcal{L}^{1}$ limit (1.4). Thus for the almost sure weak limit $\mu$ of the sequence of random measures $\mu_{k}$ we obtain the disintegration formula

[TABLE]

Our main goal is to show the existence of the limit of (1.5) in the case when $B$ is the Brownian path and to obtain a disintegration formula as in (1.9). For the purpose of application in the geometric measure theory of random intersections it is beneficial to have good control over the double intergral with respect $\mu\times\mu$.

Theorem 1.1**.**

Let $B$ be a Brownian path in $\mathbb{R}^{d}$ for $d\geq 3$ and let $\nu$ be a locally finite Borel measure on $\mathbb{R}^{d}$ such that $\nu(\{0\})=0$. Then $\nu=\nu_{R}+\nu_{\perp}$ such that there exists a Borel set $A$ such that $\nu_{\perp}(\mathbb{R}^{d}\setminus A)=0$, $P(B\cap A\neq\emptyset)=0$ and there exists a random, locally finite, Borel measure $\mu$ supported on $B$ such that

[TABLE]

and

[TABLE]

Remark 1.2*.*

Note that if $\mu$ is a random measure supported on the Brownian path $B$ and $A$ is a deterministic Borel set such that $P(B\cap A\neq\emptyset)=0$ then $\mu(A)\leq\mu(\mathbb{R}^{d}\setminus B)=0$ almost surely and so

[TABLE]

This means that there is no hope to satisfy (1.4) for $\nu_{\perp}$ in Theorem 1.1. We restate Theorem 1.1 in a stronger form of Theorem 1.22.

Our construction works for more general random closed sets than the Brownian path and in more general metric spaces than $\mathbb{R}^{d}$. We use the sum in (1.5) to define the sequence $\mu_{k}$ in the general case (see Section 1.1).

1.1 Notations

1.1.1 General assumptions throughout the paper

Definition 1.3**.**

A locally finite Borel measure $\nu$ on a topological space $X$ is called inner regular if for every Borel set $A\subseteq X$

[TABLE]

and $\nu$ is called outer regular if for every Borel set $A\subseteq X$

[TABLE]

A Hausdorff topological space $X$ is called a Radon space if every locally finite Borel measure on $X$ is outer and inner regular.

Every Polish space is a Radon space [14, Theorem 17.11] and so every locally compact, separable, metric space is a Radon space.

Throughout the paper let $(X,d)$ be a separable metric space that is also a Radon space. We notoriously use the inner regularity of Borel measures on $X$ throughout the paper. Let $\varphi:[0,\infty)\longrightarrow[0,\infty]$ be a nonnegative, continuous, monotone decreasing function with finite values on $(0,\infty)$. We consider the composition kernel $\varphi(d(x,y))$ on $X\times X$ which we denote by $\varphi(x,y)$. We note that we use $\varphi$ to denote both $\varphi(r)$ and $\varphi(x,y)$ but in the context it should be clear depending on what is the domain of $\varphi$.

Example 1.4**.**

Commonly used examples are the Riesz kernel $\varphi(x,y)=\left\|x-y\right\|^{-\alpha}$ and the logarithmic kernel $\varphi(r)=\max\{0,\log(1/r)\}$.

Let $\mathcal{Q}_{k}$ be a sequence of countable families of Borel subsets of $X$ such that $Q\cap S=\emptyset$ for $Q,S\in\mathcal{Q}_{k}$, $Q\neq S$ for all $k\in\mathbb{N}$ and

[TABLE]

where $\mathrm{diam}$ denotes the diameter in $X$. We further assume that for every $Q\in\mathcal{Q}_{k}$, $k>1$ there exists a unique $D\in\mathcal{Q}_{k-1}$ such that

[TABLE]

Define $X_{0}:=\bigcap_{k=1}^{\infty}(\bigcup_{Q\in\mathcal{Q}_{k}}Q)$.

Example 1.5**.**

Let $\mathcal{Q}_{k}=\left\{[\frac{i_{1}}{2^{k}},\frac{i_{1}+1}{2^{k}})\times\dots\times[\frac{i_{d}}{2^{k}},\frac{i_{d}+1}{2^{k}})\setminus\left\{0\right\}:i_{1},\dots,i_{d}\in\mathbb{Z}\right\}$ for $k\in\mathbb{N}$, i.e. the dyadic cubes of side length $2^{-k}$ which are left closed, right open and we exclude the origin [math] from the one that contains it. Then $X_{0}=\mathbb{R}^{d}\setminus\left\{0\right\}$.

Let $\nu$ be a finite Borel measure on $X$. Throughout most of the paper we make the following assumption in our statements that

[TABLE]

Let $(\Omega,\mathcal{A},P)$ be a probability space and $B=B_{\omega}\subseteq X$ be a random closed set ($\omega\in\Omega$) such that $\{B\cap K\neq\emptyset\}\in\mathcal{A}$ for every compact set $K\subseteq X$ and $\{B\cap Q\neq\emptyset\}\in\mathcal{A}$ for every $Q\in\mathcal{Q}_{k}$, $k\in\mathbb{N}$. We assume that

[TABLE]

for every $Q\in\mathcal{Q}_{k}$. Note that (1.13) can be obtained by discarding those $Q$ from $\mathcal{Q}_{k}$ for which $P(Q\cap B\neq\emptyset)=0$.

We write

[TABLE]

It follows from (1.12), (1.13) and that the elements of $\mathcal{Q}_{k}$ are disjoint that

[TABLE]

for every Borel set $A\subseteq X$.

1.1.2 Special assumptions for our main results

For our main results we make the assumptions of Section 1.1.2 on $\varphi$, $\mathcal{Q}_{k}$ and $B$. We list these assumptions below, however, we will note in the statement of the results and in the text when we make these assumptions.

For some $\delta>0$ there exist $c_{2},c_{3}<\infty$ such that

[TABLE]

for all $r>0$. We assume that

[TABLE]

which ensures that whenever $\nu(\{x\})>0$ for some $x\in X$ and Borel measure $\nu$ then $\iint\varphi(x,y)\mathrm{d}\nu(x)\mathrm{d}\nu(y)=\infty$. See Example 1.4.

There exists $\delta>0$ (that is the same $\delta$ as in (1.16)) and $M_{\delta}<\infty$ independent of $k$ such that for every $Q\in\mathcal{Q}_{k}$

[TABLE]

where $\mathrm{dist}(Q,S)=\inf_{x\in Q,y\in S}\left\|x-y\right\|$ and $\#A$ denotes the cardinality of $A$. There exists $0<M<\infty$, independent of $k$, such that

[TABLE]

for every $Q,S\in\mathcal{Q}_{k}$, $k\in\mathbb{N}$.

The $\varphi$-energy of a Borel measure $\nu$ on $X$ is

[TABLE]

The $\varphi$-capacity of a Borel set $K\subseteq X$ is

[TABLE]

When $\varphi(r)=r^{-\alpha}$ for some $\alpha\geq 0$ then we write $I_{\alpha}(\nu)=I_{\varphi}(\nu)$ and $C_{\alpha}(K)=C_{\varphi}(K)$.

There exists $a>0$ such that

[TABLE]

for every $Q\in\mathcal{Q}_{k}$. Hence if $C_{\varphi}(Q)>0$ for every $Q\in\mathcal{Q}_{k}$ then (1.13) holds.

There exists $0<\delta<1$ and $0<c<\infty$ such that whenever $Q\in\mathcal{Q}_{k}$, $S\in\mathcal{Q}_{n}$ and $\max\left\{\mathrm{diam}(Q),\mathrm{diam}(S)\right\}<\delta\cdot\mathrm{dist}(Q,S)$ then

[TABLE]

We further assume that $\delta>0$ is the same value for (1.16), (1.18) and (1.23). We would like to note that the assumption that $\delta<1$ is not necessary, we just use it as a convenient assumption.

Example 1.6**.**

Let $\mathcal{Q}_{k}^{i}=\left\{[\frac{i_{1}}{2^{k}},\frac{i_{1}+1}{2^{k}})\times\dots\times[\frac{i_{d}}{2^{k}},\frac{i_{d}+1}{2^{k}})\subseteq[-2^{i},2^{i})^{d}\setminus[-\frac{1}{2^{i}},\frac{1}{2^{i}})^{d}:i_{1},\dots,i_{d}\in\mathbb{Z}\right\}$ for some $k,i\in\mathbb{N}$, $i\leq k$, i.e. the dyadic cubes of side length $2^{-k}$ which are contained in $[-2^{i},2^{i})^{d}\setminus[-\frac{1}{2^{i}},\frac{1}{2^{i}})^{d}$.

We show, in Section 12.3, that for fixed $i\in\mathbb{N}$ for $\mathcal{Q}_{k}^{i}$, $(k\geq i)$ in Example 1.6 we have that (1.10), (1.11), (1.13), (1.18), (1.19), (1.22) and (1.23) hold for sufficient constants $0<\delta^{i}<1$, $M_{\delta}^{i}<\infty$, $0<M^{i}<\infty$, $a^{i}<\infty$ and $0<c^{i}<\infty$ when $B$ is the Brownian path in $\mathbb{R}^{d}$ ($d\geq 3$).

1.2 Decomposition of measures

In the spirit of (1.8) we would like to decompose $\nu$ into a vanishing part and a part for which we obtain convergence in $\mathcal{L}^{1}$. Similar results to the following proposition was published by Kahane [13, Section 3]. However, we are not aware that this kind of results appeared in the literature in English.

Proposition 1.7**.**

Let $\nu$ be a locally finite Borel measure on $X$. There exist two locally finite, Borel measures $\nu_{\varphi R}=\nu_{R}$ and $\nu_{\varphi\perp}=\nu_{\perp}$ with the following properties:

i) $\nu=\nu_{R}+\nu_{\perp}$

ii) $\nu_{R}\perp\nu_{\perp}$

iii) $\nu_{\perp}$ is singular to every locally finite Borel measure with finite $\varphi$-energy

iv) there exists a sequence of disjoint Borel sets $(A_{n})_{n\in\mathbb{N}}$ such that $\nu_{R}=\nu|_{\cup_{n\in\mathbb{N}}A_{n}}=\sum_{n\in\mathbb{N}}\nu|_{A_{n}}$ and $I_{\varphi}(\nu|_{A_{n}})<\infty$.

Notation 1.8*.*

We call $\nu_{R}$ the $\varphi$-regular part of $\nu$ and we call $\nu_{\bot}$ the $\varphi$-singular part of $\nu$. These are uniquely determined by $\nu$ and $\varphi$. When $\varphi(r)=r^{-\alpha}$ for some $\alpha\geq 0$ then we say that $\nu_{R}$ is the $\alpha$-regular part of $\nu$ and $\nu_{\bot}$ is the $\alpha$-singular part of $\nu$.

If $f:X\longrightarrow\mathbb{R}$ is a nonnegative Borel function then we denote by

[TABLE]

the measure $\tau$ defined by $\tau(A)=\intop_{A}f(x)\mathrm{d}\nu(x)$.

Proposition 1.9**.**

Let $\nu$ be a locally finite Borel measure on $X$ and let $\tau\ll\nu$ be a locally finite Borel measure on $X$. Then $\frac{\mathrm{d}\tau}{\mathrm{d}\nu}(x)\mathrm{d}\nu_{\varphi R}(x)=\mathrm{d}\tau_{\varphi R}(x)$ and $\frac{\mathrm{d}\tau}{\mathrm{d}\nu}(x)\mathrm{d}\nu_{\varphi\perp}(x)=\mathrm{d}\tau_{\varphi\perp}(x)$.

The $\varphi$-singular part $\nu_{\bot}$ of a measure is carried by a set of zero $\varphi$-capacity.

Proposition 1.10**.**

If $\nu$ is a locally finite Borel measure that is singular to every finite Borel measure with finite $\varphi$-energy then there exits a Borel set $Z\subseteq X$ such that $\nu(X\setminus Z)=0$ and $C_{\varphi}(Z)=0$.

We prove Proposition 1.7, Proposition 1.9 and Proposition 1.10 in Section 4.

1.3 Summary of the main results

Let $\mu$ and $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$ (for the definition of random, finite, Borel measures see Definition 3.1). We say that $\mu_{k}$* weakly converges to $\mu$ subsequentially in probability* if for every subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ there exists a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ and an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\mu_{\beta_{k}}$ converges weakly to $\mu$ on the event $H$.

Let $\mu$ and $\mu_{k}$ be a sequence of random, locally finite, Borel measures on $X$ (for the definition of random, locally finite, Borel measures see Definition 3.27). We say that $\mu_{k}$* vaguely converges to $\mu$ in probability* if $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to $\intop_{X}f(x)\mathrm{d}\mu(x)$ in probability for every compactly supported continuous function $f$.

For a Borel measure $\nu$ on $X$ let $\mathrm{supp}(\nu)$ denote the support of $\nu$, that is the smallest closed set with full measure, i.e.

[TABLE]

Below we define the main object of the paper.

Definition 1.11**.**

Let $\nu$ be a finite, Borel measure on $X$. If for every finite, Borel measure $\tau\ll\nu$ there exists a random, finite, Borel measure $\mathcal{C}(\tau)$ that satisfies the following:

i.) $\mathcal{C}_{k}(\tau)$ weakly converges to $\mathcal{C}(\tau)$ subsequentially in probability,

ii.) the sequence $\intop_{X}f(x)\mathrm{d}\mathcal{C}_{k}(\tau)(x)$ converges to a random variable $S_{\tau}(f)$ in probability with $E(\intop_{X}f(x)\mathrm{d}\mathcal{C}(\tau)(x))=E(S_{\tau}(f))=\intop_{X}f(x)\mathrm{d}\tau_{R}(x)$ for every $f:X\longrightarrow\mathbb{R}$ Borel measurable function such that $\intop_{X}\left|f(x)\right|\mathrm{d}\tau(x)<\infty$,

iii.) for every countable collection of deterministic Borel measurable functions $f_{n}:X\longrightarrow\mathbb{R}$ with $\intop_{X}\left|f_{n}(x)\right|\mathrm{d}\tau(x)<\infty$ we have that $\intop_{X}f_{n}(x)\mathrm{d}\mathcal{C}(\tau)(x)=S_{\tau}(f_{n})$ for every $n\in\mathbb{N}$ almost surely,

iv.) for every countable collection of deterministic, Borel sets $A_{n}\subseteq X$ with $\tau(A_{n})<\infty$ we have that $S_{\tau}(\chi_{A_{n}})=\mathcal{C}(\tau)(A_{n})$ for every $n\in\mathbb{N}$ almost surely,

v.) $E(\mathcal{C}(\tau)(A))=E(S_{\tau}(\chi_{A}))=\tau_{R}(A)\leq\tau(A)$ for every Borel set $A\subseteq X$ with $\tau(A)<\infty$,

vi.) $\mathcal{C}_{k}(\tau_{R})(A)$ converges to $\mathcal{C}(\tau)(A)$ in $\mathcal{L}^{1}$ for every Borel set $A\subseteq X$ with $\tau(A)<\infty$

vii.) $\mathcal{C}(\tau_{\bot})=0$ almost surely,

viii.) $\mathcal{C}(\tau)=\mathcal{C}(\tau_{R})$ almost surely,

ix.) if $\tau=\sum_{i=1}^{\infty}\tau^{i}$ for a sequence of finite, Borel measures $\tau^{i}$ then $\mathcal{C}(\tau)=\sum_{i=1}^{\infty}\mathcal{C}(\tau^{i})$ almost surely,

x.) if $f:X\longrightarrow\mathbb{R}$ is a nonnegative Borel function such that $\intop_{X}f(x)\mathrm{d}\tau(x)<\infty$ then $\mathcal{C}(f(x)\mathrm{d}\tau(x))=f(x)\mathrm{d}\mathcal{C}(\tau)(x)$, in particular, if $\gamma\in[0,\infty)$ then $\mathcal{C}(\gamma\cdot\tau)=\gamma\cdot\mathcal{C}(\tau)$ almost surely,

xi.) $\mathrm{supp}\mathcal{C}(\tau)\subseteq\mathrm{supp}\tau\cap B$ almost surely,

then we say that the conditional measure of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ and it is* $\mathcal{C}(\nu)$*.

If $X$ is locally compact and $\nu$ is a locally finite, Borel measure on $X$ and for every locally finite, Borel measure $\tau\ll\nu$ there exists a random, locally finite, Borel measure $\mathcal{C}(\tau)$ that satisfies ii.)-viii.), xi.) and additionally also satisfies the following:

i.)* $\mathcal{C}_{k}(\tau)$ vaguely converges to $\mathcal{C}(\tau)$ in probability,

ix.)* if $\tau=\sum_{i=1}^{\infty}\tau^{i}$ for a sequence of locally finite, Borel measures $\tau^{i}$ then $\mathcal{C}(\tau)=\sum_{i=1}^{\infty}\mathcal{C}(\tau^{i})$ almost surely,

x.)* if $f:X\longrightarrow\mathbb{R}$ is a nonnegative Borel function such that for every $y\in X$ there exists a neigbourhood $U$ of $y$ such that $\intop_{U}f(x)\mathrm{d}\tau(x)<\infty$ then $\mathcal{C}(f(x)\mathrm{d}\tau(x))=f(x)\mathrm{d}\mathcal{C}(\tau)(x)$, in particular, if $\gamma\in[0,\infty)$ then $\mathcal{C}(\gamma\cdot\tau)=\gamma\cdot\mathcal{C}(\tau)$ almost surely,

then we say that the conditional measure of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ and it is* $\mathcal{C}(\nu)$*.

Remark 1.12*.*

We would like to note that when we say that the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ some may think it is a bit incorrect because we actually mean that $\mathcal{C}(\tau)$ exists for every $\tau\ll\nu$. It follows from the definition that if $\mathcal{C}(\nu)$ exists then $\mathcal{C}(\tau)$ also exists for every $\tau\ll\nu$. However, there is a characterisation whether $\mathcal{C}(\nu)$ exists which need no information about any $\tau\ll\nu$, see Theorem 1.16. So we simply say that $\mathcal{C}(\nu)$ exists when we do mean that for every $\tau\ll\nu$ the Properties in Definition 1.11 are all satisfied. Throughout the paper when we write $S(f)$ for a function $f$ and we do not indicate otherwise in lower case then we mean $S_{\nu}(f)$ by $S(f)$.

Remark 1.13*.*

By Property *ii.) *and iv.) of Definition 1.11 it follows that for every countable collection of Borel sets $A_{n}$ we have that $\mathcal{C}_{k}(\nu)(A_{n})$ converges to $\mathcal{C}(\nu)(A_{n})$ in probability as $k$ goes to $\infty$. We note that in general it is not true that $\mathcal{C}(\nu)(A)$ is the limit of $\mathcal{C}_{k}(\nu)(A)$ in probability for every Borel set $A$ simultaneously. Let $\nu=\lambda$ and $A$ be a set of [math] Lebesgue measure. Then $\mathcal{C}_{k}(\nu)(A)=0$ for every $k$. However, $\mathcal{C}(\nu)(A)=0$ cannot always hold for every such $A$ simultaneously because usually $\mathcal{C}(\nu)$ is supported on a set of [math] Lebesgue measure by Property xi.) of Definition 1.11. For instance, in the case of $B$ being the Brownian path.

We defined the conditional measure of finite Borel measures and in locally compact spaces we defined the conditional measure of locally finite Borel measures. To avoid confusion we need to show that the two definitions of the conditional measure of finite measures in locally compact spaces are the same. Proposition 1.14 is proven at the end of Section 3.4.

Proposition 1.14**.**

If $X$ is locally compact and $\nu$ is a finite Borel measure then the two definitions of the conditional measure of $\nu$ on $B$ with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ in Definition 1.11 are equivalent and the limits $\mathcal{C}(\nu)$ in Property i.) and in Property i.) are the same almost surely.*

We develop the theory of weak convergence subsequentially in probability and vague convergence in probability in Section 3, that is essential in showing the existence of the conditional measure. The proofs are based on classical functional analysis and classical probability. The following proposition provides a tool to show the existence of the conditional measure.

Proposition 1.15**.**

Let $\nu$ be a deterministic, finite, Borel measure on $X$. Let $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$ such that $\mu_{k}\ll\nu$ almost surely for every $k$, there exists $c:X\longrightarrow\mathbb{R}$ such that $\int c(x)\mathrm{d}\nu(x)<\infty$ and $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c(x)$ for $\nu$ almost every $x\in X$ for every $k$. Assume that $\mu_{k}(A)$ converges to a random variable $\mu_{\infty}(A)$ in probability for every compact set $A\subseteq X$. Then $\mu_{k}$ weakly converges to a random, finite, Borel measure $\mu$ subsequentially in probability.

Proposition 1.15 also extends to locally finite measures when $X$ is locally compact and we consider the vague convergence in probability. We prove these results in Section 3. The notion of weak convergence subsequentially in probability was considered independently by Berestycki in [1, Section 6] in the context of Gaussian multiplicative chaos, the existence of a random measure in that paper depends on a similar result to Proposition 1.15, though it is not explicitly stated. The following theorem gives a characterisation when $\mathcal{C}(\nu)$ exists and is proved in Section 3.4.

Theorem 1.16**.**

Let $\nu$ be a finite, Borel measure on $X$ or let $X$ be locally compact and $\nu$ be a locally finite, Borel measure on $X$. Assume that $\nu(X\setminus X_{0})=0$. Then the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ if and only if $\mathcal{C}_{k}(\nu|_{D})(X)=\mathcal{C}_{k}(\nu)(D)$ converges in $\mathcal{L}^{1}$ for every compact set $D\subseteq X_{0}$ with $I_{\varphi}(\nu|_{D})<\infty$ and $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability for every compact set $D\subseteq X_{0}$.

The sequence $\mathcal{C}_{k}(\nu)$ might give the impression of a $T$-martingale that was introduced by Kahane [12]. However, when $B$ is the Brownian path $\mathcal{C}_{k}(\nu)$ is not a $T$-martingale with respect to the natural filtration $\mathcal{F}_{k}=\sigma\{B\cap Q\neq\emptyset\}_{Q\in\mathcal{Q}_{k}}$ and we cannot prove almost sure convergence of $\mathcal{C}_{k}(\nu)$, that is why we needed to develop the convergence of random measures in probability. Despite that, $\mathcal{C}_{k}(\nu)$ and its limit $\mathcal{C}(\nu)$ if the conditional measure exists, exhibits many similar properties to Kahane‘s $T$-martingales. To get around the trouble that $\mathcal{C}_{k}(\nu)$ is not necessarily a $T$-martingale we define the following kernels that are key objects in showing that the limit of $\mathcal{C}_{k}(\nu)$ exists.

Notation 1.17*.*

For $x\in X$ and $k\in\mathbb{N}$ let $Q_{k}(x)=Q$ if $x\in Q$ for some $Q\in\mathcal{Q}_{k}$ and $Q_{k}(x)=\emptyset$ otherwise. There is at most one such $Q$ since elements of $\mathcal{Q}_{k}$ are disjoint hence $Q_{k}(x)$ is well-defined.

Definition 1.18**.**

For $k,n\in\mathbb{N}$ let $F_{k,n}:X\times X\longrightarrow\mathbb{R}$ be the nonnegative function

[TABLE]

Definition 1.19**.**

We define the following functions

[TABLE]

and

[TABLE]

and their limits

[TABLE]

and

[TABLE]

Remark 1.20*.*

If (1.23) holds for some $\delta>0$ then

[TABLE]

for $x\neq y$.

In Section 5 we prove that, if $C_{\varphi}(D)=0$ for some compact set $D\subseteq X_{0}$ that implies that $B\cap D=\emptyset$ almost surely, then $\mathcal{C}_{k}(\nu_{\bot})$ converges to [math] weakly subsequentially in probability. If $X$ is locally compact then even for locally finite $\nu$ it follows that $\mathcal{C}_{k}(\nu_{\bot})$ converges to [math] vaguely in probability. Throughout the paper we assume that $\nu(X\setminus X_{0})=0$ because when $\nu(X_{0})=0$ then $\mathcal{C}_{k}(\nu)$ converges to [math] (see Section 5). The existence of the kernel $F(x,y)=\underline{F}(x,y)=\overline{F}(x,y)$ is the key assumption in order to prove the existence of the conditional measure in the nondegenerate case. Using the observation that $E\left(\mathcal{C}_{k}(\nu)(A)^{2}\right)=\intop_{A}\intop_{A}F_{k,k}(x,y)\mathrm{d}\nu(x)\mathrm{d}\nu(y)$, we show, throughout Section 6 and 7, that if $I_{\varphi}(\nu)<\infty$ then $\mathcal{C}_{k}(\nu)$ converges in $\mathcal{L}^{2}$ and so in $\mathcal{L}^{1}$. Finally, using the measure decomposition result of Proposition 1.7 we conclude one of our deepest result in Section 8 and Section 9. Theorem 1.21 is proven at the end of Section 9.

Theorem 1.21**.**

Assume that (1.17), (1.19) and (1.22) hold and there exists $0<\delta<1$ such that (1.16), (1.23) and (1.18) hold. Assume that $F(x,y)=\underline{F}(x,y)=\overline{F}(x,y)$ for every $(x,y)\in X\times X$. Assume that if $C_{\varphi}(D)=0$ for some compact set $D\subseteq X_{0}$ then $B\cap D=\emptyset$ almost surely. Let either $\nu$ and $\tau$ be finite Borel measures on $X$ or $X$ be locally compact and $\nu$ and $\tau$ be locally finite Borel measures on $X$. Assume that $\nu(X\setminus X_{0})=0$ and $\tau(X\setminus X_{0})=0$. Then the conditional measure $\mathcal{C}(\nu)$ of $\nu$ and $\mathcal{C}(\tau)$ of $\tau$ on $B$ exist with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ and

[TABLE]

for every $f:X\times X\longrightarrow\mathbb{R}$ Borel function with $\intop\intop F(x,y)\left|f(x,y)\right|\mathrm{d}\nu_{R}(x)\mathrm{d}\tau_{R}(y)<\infty$, in particular if $\intop_{X}\intop_{X}\varphi(x,y)\left|f(x,y)\right|\mathrm{d}\nu(x)\mathrm{d}\tau(y)<\infty$.

We study the conditional measure on the Brownian path in Section 12. We show that the conditions of Theorem 1.21 hold for the Brownian path $B\subseteq\mathbb{R}^{d}$ ($d\geq 3$) and the sequence $\mathcal{Q}_{k}^{i}$, $(k\geq i)$ in Example 1.6 for sufficient constants that depends on $i$ and $d$. However, our main goal is to show the existence of the conditional measure with respect to $\mathcal{Q}_{k}$, $(k\geq 1)$ in Example 1.5. In Section 10 we discuss how we can extend Theorem 1.21 to a $\mathcal{Q}_{k}$, $(k\geq 1)$ if we know that that conditions of Theorem 1.21 hold for $\mathcal{Q}_{k}^{i}$, $(k\geq i)$ such that $\mathcal{Q}_{k}^{i}\subseteq\mathcal{Q}_{k}$ and $\mathcal{Q}_{k}^{i}$ is approaching $\mathcal{Q}_{k}$ in some sense as $i$ goes to $\infty$. For the exact statement see Theorem 10.4. Applying this result to the Brownian path we conclude in Section 12.4 the following theorem.

Theorem 1.22**.**

Let $B$ be a Brownian path in $\mathbb{R}^{d}$ for $d\geq 3$. Let $\nu$ be a locally finite Borel measure on $\mathbb{R}^{d}$ such that $\nu(\{0\})=0$. Then the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) (where $\mathcal{Q}_{k}$ is as in Example 1.5) with regularity kernel $\varphi(x,y)=\left\|x-y\right\|^{2-d}$. Let $\tau$ also be a locally finite Borel measure on $\mathbb{R}^{d}$ such that $\tau(\{0\})=0$. Then

[TABLE]

for every Borel function $f(x,y):X\times X\longrightarrow\mathbb{R}$ with $\intop\intop\frac{\left\|x\right\|^{d-2}+\left\|y\right\|^{d-2}}{\left\|x-y\right\|^{d-2}}\left|f(x,y)\right|\mathrm{d}\nu_{R}(x)\mathrm{d}\tau_{R}(y)<\infty$.

Recall, that in (1.25) $\nu_{R}$ and $\tau_{R}$ refers to the $(d-2)$-regular part of the measures (see Notation 1.8). Note that if $\nu$ is a point mass on $\{0\}$ and we consider the $\mathcal{Q}_{k}$ of Example 1.5 with the difference that we do not remove the origin from the box containing it then $\mathcal{C}_{k}(\nu)=\nu$ for every $n$ because the Brownian path always intersects the cube containing [math]. In that case $\nu$ is almost surely supported on $B$ hence the deterministic $\mu=\nu$ itself solves the equation $\mathrm{d}\mu\mathrm{d}P=\mathrm{d}\nu$ with $\mu$ is supported on $B$ almost surely. However, (1.25) is not satisfied. Note that in the natural decomposition $\nu=\nu_{R}+\nu_{\bot}$ the point mass on $\{0\}$ belongs to the singular part, although that is the only singular measure for which the problem can be solved. It is because any singular measure with $\nu_{\bot}(\{0\})=0$ is carried by a set that is almost surely not intersected by $B$ (see Remark 1.2).

Remark 1.23*.*

Let $\nu$ be a finite Borel measure such that $\mathrm{supp}\nu$ is bounded away from [math] and infinity and $I_{\beta}(\nu)<\infty$ for some $\beta>d-2$. One consequence of the double integration formula (1.25) that $I_{\beta+2-d}(\mathcal{C}(\nu))<\infty$ almost surely. This is a useful tool in the geometric measure theory of the random intersection $B\cap K$ for some fixed deterministic Borel set $K$.

The question naturally rises what happens when $\nu$ is the Lebesgue measure. The answer is given in terms of the occupation measure of the Brownian motion, that counts the amount of time the Brownian motion spends inside a set. Section 13 is dedicated to deal with this question.

Theorem 1.24**.**

Let $B_{0}(t)$ be a Brownian motion in $\mathbb{R}^{d}$ for $d\geq 3$, let $B$ be the range of the Brownian motion and let $\lambda$ be the Lebesgue measure in $\mathbb{R}^{d}$. Let

[TABLE]

be the occupation measure of $B_{0}$. Then

[TABLE]

and

[TABLE]

almost surely where

[TABLE]

and $\Gamma(x)=\intop_{0}^{\infty}s^{x-1}e^{-s}\mathrm{d}s$ is the Euler’s Gamma function.

This result gives us a tool to calculate the occupation measure, $\tau(A)=c(d)\intop_{A}\left\|x\right\|^{2-d}\mathrm{d}\mathcal{C}(\lambda)(x)$ can be approximated by $c(d)\intop_{A}\left\|x\right\|^{2-d}\mathrm{d}\mathcal{C}_{k}(\lambda)(x)$ which converges to $\tau(A)$ in probability. To calculate the value of $c(d)\intop_{A}\left\|x\right\|^{2-d}\mathrm{d}\mathcal{C}_{k}(\lambda)(x)$ we only need to know which boxes of $\mathcal{Q}_{k}$ does $B$ intersect. In Section 13, while we prove Theorem 1.24, we basically show that

[TABLE]

converges to $\tau(K)$ in probability for every deterministic compact set $K$, where $\varphi(x,y)=\left\|x-y\right\|^{2-d}$ (note that $\alpha=1/C_{\varphi}([0,1]^{d})$ in Section 13 by Remark 13.20). These results also show that the occupation measure only depends on the trajectory of the Brownian path, we need not to know the parametrisation, however, this fact also follows from a result of Ciesielski and Taylor [4, Theorem 5] which states that $\tau(K)$ equals to a deterministic constant (depending on $d$) times the Hausdorff measure of $K\cap B$ with the gauge function $x^{2}\log\log x^{-1}$. Note that the expression in (1.27) differs from the definition of the dyadic net measure with gauge function $x^{2}$ in that the cubes are equal size rather than arbitrary. Since the net measure is equivalent to the Hausdorff measure, the expression in (1.27) cannot converge to the net measure because that would contradict the Ciesielski-Taylor result.

Remark 1.25*.*

The Green‘s function of the Brownian motion in $\mathbb{R}^{d}$ is $G(x,y)=c(d)\left\|x-y\right\|^{2-d}$ for the constant $c(d)$ in (1.26), see [19, Theorem 3.33].

The following proposition is known on the first moment of the occupation measure, see [19, Theorem 3.32].

Proposition 1.26**.**

For every Borel set $A\subseteq\mathbb{R}^{d}$ ($d\geq 3$)

[TABLE]

The following theorem calculates the second moment of the occupation measure.

Theorem 1.27**.**

We have that

[TABLE]

for every Borel function $f:\mathbb{R}^{d}\times\mathbb{R}^{d}\longrightarrow\mathbb{R}$ such that $\intop\intop\left|f(x,y)\right|\frac{\left\|x\right\|^{d-2}+\left\|y\right\|^{d-2}}{\left\|x-y\right\|^{d-2}\cdot\left\|x\right\|^{d-2}\cdot\left\|y\right\|^{d-2}}\mathrm{d}x\mathrm{d}y<\infty$.

Theorem 1.27 follows from Theorem 1.24 and Theorem 1.22. As an application of Theorem 1.27, we obtain a formula for the second moment of the occupation measure

[TABLE]

We note however, that (1.28) could be deduced, via a direct calculation, from the transition probability kernels $p^{*}(t,x,y)$. For the definition of $p^{*}(t,x,y)$ see [19, Theorem 3.30]. Alternatively, (1.28) can also be deduced from Kac’s moment formula (see [11]).

For a wide class of random closed sets we have that the hitting probability of a compact set $K$ is comparable to the $\varphi$-capacity of $K$ for a sufficient kernel $\varphi$. See Proposition 14.2 in case of the ‘percolation limit set’ or Proposition 1.30 in case of the Brownian path. In Section 11 we discuss the probability of the nonextinction of the conditional measure and we establish analogous results to the hitting probabilities, namely the probability of the nonextinction of $\mathcal{C}(\nu)$ is comparable to the capacity of the measure $\nu$. For the definition of capacities of measures $C_{\varphi}(\nu)$ and $\overline{C_{\varphi}}(\nu)$ see Definition 11.1 and Definition 11.2.

Theorem 1.28**.**

Assume that the conditions of Theorem 1.21 hold and $P(D\cap B\neq\emptyset)\leq b\cdot C_{\varphi}(D)$ for every compact set $D\subseteq X_{0}$. Let $\nu$ be a finite Borel measure such that $\nu(X\setminus X_{0})=0$. Then

[TABLE]

We prove Theorem 1.28 in Section 11 and the conclusion can be extended for locally finite measures when $X$ is locally compact (see Remark 11.8). The following result gives lower and upper bound for the probability of nonextinction of $\mathcal{C}(\nu)$ when $B$ is the Brownian path, that we show in Section 12.5.

Theorem 1.29**.**

Let $B$ be a Brownian path in $\mathbb{R}^{d}$ for $d\geq 3$. Let $\nu$ be a locally finite Borel measure on $\mathbb{R}^{d}$ such that $\nu(\{0\})=0$. Let $\mathcal{C}(\nu)$ be the conditional measure of $\nu$ on $B$ with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) ($\mathcal{Q}_{k}$ is as in Example 1.5) with regularity kernel $\varphi(x,y)=\left\|x-y\right\|^{2-d}$. Then

[TABLE]

for

[TABLE]

The results on the probability of nonextinction suggest an analogy between the random intersection $B\cap K$ for a fixed compact set $K$ and the conditional measure $\mathcal{C}(\nu)$ of a fixed measure $\nu$ on $B$. Compare Theorem 1.29 to the following known result on hitting probabilities.

Proposition 1.30**.**

Let $B$ be a Brownian path in $\mathbb{R}^{d}$ for $d\geq 3$. Let $A\subseteq\mathbb{R}^{d}\setminus\{0\}$ be a compact set. Then

[TABLE]

where $F$ is as in (1.29).

Let $G(x,y)=c(d)\left\|x-y\right\|^{2-d}$ be the Green‘s function of the Brownian motion in $\mathbb{R}^{d}$ (see [19, Theorem 3.33]) and let $M(x,y)=G(x,y)/G(0,y)$ be the Martin‘s kernel. Then $F(x,y)=M(x,y)+M(y,x)$ and so $2I_{M}(\nu)=I_{F}(\nu)$ for every finite Borel measure $\nu$. Thus Proposition 1.30 is a reformulation of [19, Theorem 8.24].

We stated many of our main results in general separable metric Radon space $X$ for a ‘reasonable’ random closed set $B$. In Section 14 we let $X=\partial T$ to be the boundary of an infinite rooted tree $T$. We discuss the theory of the conditional measure when $B$ is a ‘percolation limit set’. We establish many properties of the conditional measure such as the double integration formula and the probability of nonextinction. In the end, we prove that $\mathcal{C}(\nu)$ is a certain ‘random multiplicative cascade measure’. For the exact statement of these results and the discussion see Section 14. Section 14 is complete without the knowledge of the sections on the Brownian motion, i.e. does not rely on Section 10, Section 12 and Section 13.

We hope that our methods can be extended to a rich family of natural random closed sets, however, this study is already extensive but the question can be a subject of further investigation. The key to make the machinery work is the existence of $F(x,y)$, the rest of the assumptions in Section 1.1.2 are more straightaway if we have $F(x,y)$. In the case of the two dimensional Brownian motion to obtain a closed set, we need to stop the Brownian motion at some point. A natural candidate is the two dimensional Brownian motion that is stopped at a random exponential time independent of the Brownian motion, and let $B$ to be the range of the motion till that stopped time. We suspect that the assumptions of Theorem 1.21 also hold locally in the two dimensional case and due to Section 10 can be extended globally to the plane. The proof shall differ a bit due to the fact that the Geen’s function is merely different in two dimension than in higher dimensions. Also other random processes might satisfy our assumptions, the list could include processes like $\alpha$-stable Levy processes, the fractional Brownian motion or the Mandelbrot percolation set.

We note that all our results can be extended to finite signed and complex measures or if $X$ is locally compact then to locally finite signed and complex measures. It can be seen by applying our results to the measure parts in the decomposition of signed and complex measures.

1.4 Application

We introduced the concept of the conditional measure on a random set by the analogy of slicing measures (1.2) by straight lines. For the conditional measure we wish to solve the problem that we require a random measure $\mu$ to be supported on $B$ and $\mathrm{d}\mu\mathrm{d}P=\mathrm{d}\nu$ for a given measure $\nu$. As we discussed in Remark 1.2 it is not possible to solve the problem for $\nu_{\bot}$ in the case of the Brownian path and we can solve it for $\nu_{R}$. However, the solution is not unique, one can take a random variable $Y$ independent of $B$ that takes value $p^{-1}$ with probability $p$ and value [math] with probability $1-p$. Then if $\mu$ is a solution then $Y\mu$ is also a solution. Even if we require $\mu$ to be measurable with respect to the $\sigma$-algebra $\mathcal{F}$ generated by the events $\{Q\cap B\neq\emptyset\}$ where $Q\in\mathcal{Q}_{k}$, $k\in\mathbb{N}$ the solution is not unique. One can construct a random point mass $\mu$, such that $\mu$ is $\mathcal{F}$-measurable and $\mathrm{d}\mu\mathrm{d}P=\mathrm{d}\lambda$. However, a disadvantage of the point mass that it does not behave well for the double integration formula (1.25). The construction of $\mathcal{C}(\nu)$ defined as the limit of $\mathcal{C}_{k}(\nu)$ is in some sense a random measure that is nicely spread on $B$ unlike a point mass. The construction is analogous to the slice measures (1.1). In the application of the slice measures a weaker double integration formula [17, Theorem 10.7] provides a strong tool for the geometric measure theory of slices. Similarly the double integration formula for $\mathcal{C}(\nu)$ gives us a powerful weapon and can be applied to analogous problems that the slice measures are used for in the case of intersections with straight lines. We outline below one further application that we believe is merely new.

It can be deduced from Remark 1.23 that

[TABLE]

conditional on $\mathcal{C}(\nu)(X)>0$, where $\dim_{H}$ denotes the Hausdorff dimension. Let $K\subseteq\mathbb{R}^{d}\setminus\{0\}$ be a fixed compact set. Assume that $\nu$ is a finite deterministic measure on $K$ such that

[TABLE]

Then $\dim_{H}\nu-(d-2)\leq\dim_{H}K\cap B$ almost surely conditional on $K\cap B\neq\emptyset$ by (1.31), (1.32) and Property xi.) of Definition 1.11. We suspect that (1.32) holds for the harmonic measure of $K$ for the Brownian motion and for the equilibrium measure of $K$ for the capacity kernel $\left\|x-y\right\|^{2-d}$. An essential almost sure upper bound for $\dim_{H}K\cap B$ has been known for a long time [9, 10], namely $\left\|\dim_{H}K\cap B\right\|_{\infty}=\max\{0,\dim_{H}K-(d-2)$, however we are not aware of a general method to give almost sure lower bounds on $\dim_{H}K\cap B$ conditional on intersection.

We note that the harmonic measure does not necessarily give the essential lower bound. If $K$ is a ball in $\mathbb{R}^{d}$ not containing the origin then the Harmonic measure is supported on $\partial K$ which gives lower estimate of dimension $1$ while the correct dimension of the intersection is almost surely $2$ conditional on intersection. For $K$ being the ball the Lebesgue measure is the ideal choice for $\nu$.

In general, if $\nu_{n}$ is a sequence of measures supported on $K$ such that $\lim_{n\rightarrow\infty}P(\mathcal{C}(\nu_{n})(X)>0)=P(K\cap B\neq\emptyset)$ then $\limsup_{n\rightarrow\infty}\dim_{H}\nu_{n}-(d-2)\leq\dim_{H}K\cap B$ almost surely conditional on $K\cap B\neq\emptyset$. This means that we can provide almost sure lower bound even if we cannot find the ideal measure that satisfies (1.32) but we can find a sequence of measures that in limes satisfies (1.32).

2 Preliminary remarks

In this section we summarise the background and preliminary lemmas.

For $x\in X$, $r>0$ let $B(x,r)=\left\{y\in X:d(x,y)<r\right\}$ and for a set $A\subseteq X$ let $B(A,r)=\left\{y\in X:\exists x\in A,d(x,y)<r\right\}$.

Notation 2.1*.*

For $A\subseteq X$ let

[TABLE]

be the characteristic function of $A$.

Notation 2.2*.*

For a probability event $A\in\mathcal{A}$ let

[TABLE]

be the* indicator function of $A$*.

2.1 Convergence in probability

We list below some folklore properties of the convergence in probability that we need throughout the paper.

Definition 2.3**.**

Let $Y_{n}$ ($n\in\mathbb{N}$) and $Y$ be random variables. We say that $Y_{n}$* converges to $Y$ in probability* (as $n$ goes to $\infty$) if

[TABLE]

for every $\varepsilon>0$. If the random variables take values in a metric space than we can replace the difference $\left|.\right|$ by the metric to obtain a definition of convergence in probability for random variables that take values in a metric space.

Remark 2.4*.*

Let $Y$ and $Y_{n}$ be a sequence of random variables that take values in a metric space. Then $Y_{n}$ converges to $Y$ in probability if and only if for every subsequence $\left\{n_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ there exists a subsequence $\left\{j_{k}\right\}_{k=1}^{\infty}$ of $\left\{n_{k}\right\}_{k=1}^{\infty}$ such that $Y_{j_{k}}$ converges to $Y$ almost surely (see [8, Theorem 2.3.2]).

Remark 2.5*.*

The convergence in probability is a metric convergence and is completely metrizable by the following metric $\rho$ (see [8, Exercise 2.3.8 and 2.3.9]). For random variables $Y,Z$ let

[TABLE]

Remark 2.6*.*

Note that $\rho(Y,Z)\leq E\left(\left|Y-Z\right|\right)$. We use this fact without reference throughout the paper.

Lemma 2.7**.**

Let $Y_{n}:\Omega\longrightarrow M$ be a sequence of random variables that take values in a metric space $M$. If $Y:\Omega\longrightarrow M$ is a function and there exists an event $H$ with $P(H)=1$ such that $Y_{n}(\omega)$ converges to $Y(\omega)$ for every $\omega\in H$ then $Y$ is a random variable.

For the proof of Lemma 2.7 see [6, Theorem 4.2.2]

Lemma 2.8**.**

Assume that $\left\{f_{k}^{i}\right\}_{i,k\in\mathbb{N}}$ is a family of random variables such that $f_{k}^{i}$ converges in probability as $k$ goes to infinity for every $i$. Then for every subsequence $\left\{n_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ there exists a subsequence $\left\{j_{k}\right\}_{k=1}^{\infty}$ of $\left\{n_{k}\right\}_{k=1}^{\infty}$and there exists a probability event $H\in\mathcal{A}$ with $P(H)=1$ such that $f_{j_{k}}^{i}$ converges on the event $H$ as $k$ goes to infinity for every $i$.

Proof.

Let $\left\{\alpha_{k}^{1}\right\}_{k=1}^{\infty}$ be a subsequence of $\left\{n_{k}\right\}_{k=1}^{\infty}$ such that $f_{\alpha_{k}^{1}}^{1}$ converges almost surely and let $j_{1}=\alpha_{1}^{1}$. If $\left\{\alpha_{k}^{i}\right\}_{k=i}^{\infty}$ and $j_{1},\dots j_{i}$ are defined let $\left\{\alpha_{k}^{i+1}\right\}_{k=i+1}^{\infty}$ be a subsequence of $\left\{\alpha_{k}^{i}\right\}_{k=i+1}^{\infty}$ such that $f_{\alpha_{k}^{i+1}}^{i+1}$ converges almost surely as $k$ goes to infinity and let $j_{i+1}=\alpha_{i+1}^{i+1}$. Then $\left\{j_{k}\right\}_{k=i}^{\infty}$ is a subsequence of $\left\{\alpha_{k}^{i}\right\}_{k=i}^{\infty}$ and hence $f_{j_{k}}^{i}$ converges almost surely as $k$ goes to infinity for every $i$. ∎

Lemma 2.9**.**

Let $Y$ and $Y_{n}$ ($n\in\mathbb{N}$) be a sequence of random variables. If for every $\varepsilon>0$ and every subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ we can find a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ such that

[TABLE]

then $Y_{n}$ converges to $Y$ in probability as $n$ goes to $\infty$.

Proof.

In a topological space $x_{k}$ converges to $x$ if and only if for every subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ we can find a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ such that $x_{\beta_{k}}$ converges to $x$. Applying this to $x_{k}=P\left(\left|Y_{k}-Y\right|>\varepsilon\right)$ it follows that $\lim_{k\rightarrow\infty}P\left(\left|Y_{k}-Y\right|>\varepsilon\right)=0$. ∎

Lemma 2.10**.**

Let $Y_{n}$ be a sequence of real valued random variables such that $Y_{n}$ converges to $Y$ in probability and $\lim_{n\rightarrow\infty}E\left|Y_{n}\right|=E\left|Y\right|<\infty$. Then $Y_{n}$ converges to $Y$ in $\mathcal{L}^{1}$.

For details of the proof see [8, Theorem 5.5.2]

Lemma 2.11**.**

If $Y_{n}$ ($n\in\mathbb{N}$) is a sequence of nonnegative, real valued random variables, $Y_{n}$ converges to $Y$ in probability and there exists $c<\infty$ such that $E(Y_{n})\leq c$ for every $n\in\mathbb{N}$ then $E(Y)\leq c$.

Proof.

Let $n_{k}$ be a sequence such that $Y_{n_{k}}$ converges to $Y$ almost surely. Then by Fatou‘s lemma

[TABLE]

∎

2.2 Weak*∗* and vague convergence of measures

We recall some properties of the weak*∗* and vague convergences of measures and some related lemmas.

Let $C_{b}(X)$ denote the *space of bounded continuous functions on $X$ *equipped with the supremum norm and let $C_{c}(X)$ denote the *space of all compactly supported continuous functions on $X$ *equipped with the supremum norm. We denote by $\mathrm{supp}(f)$ the support of a function $f:X\longrightarrow\mathbb{R}$.

Lemma 2.12**.**

Let $\nu$ be a finite, Borel measure on $X$ and $A\subseteq X$ be a Borel set. Then there exists a sequence of disjoint compact subsets $K_{1},K_{2},\dots$ of $A$ such that $\nu(A\setminus\cup_{i=1}^{\infty}K_{i})=0$.

Remark 2.13*.*

Lemma 2.12 easily follows from the inner regularity of $\nu$. The statement also holds for a locally finite Borel measure $\nu$. It can be deduced from that $\nu$ is locally finite and $X$ satisfies the Lindelöf property as it is a separable metric space.

Definition 2.14**.**

Let $\mu$ and $\mu_{k}$ ($k\in\mathbb{N}$) be finite Borel measures on $X$. We say that $\mu_{k}$* weakly converges to $\mu$* (as $k$ goes to $\infty$) if

[TABLE]

for every $f\in C_{b}(X)$.

Remark 2.15*.*

It is well-known, that $\mu_{k}$ converges to $\mu$ weakly if and only if $\mu(G)\leq\liminf_{k\rightarrow\infty}\mu_{k}(G)$ for every open set $G$. See [2, Theorem 2.1].

Lemma 2.16**.**

Let $\mu$ and $\nu$ be finite Borel measures on $X$ such that $\mu\neq\nu$. Then there exists $f\in C_{b}(X)$ such that $\intop_{X}f(x)\mathrm{d}\mu(x)\neq\intop_{X}f(x)\mathrm{d}\nu(x)$.

Lemma 2.16 is shown in [2, Theorem 1.2]. In locally compact space the same proof results the following lemma because by inner regularity the measure of compact sets determines the measure.

Lemma 2.17**.**

Let $X$ be locally compact. Let $\mu$ and $\nu$ be locally finite Borel measures on $X$ such that $\mu\neq\nu$. Then there exists $f\in C_{c}(X)$ such that $\intop_{X}f(x)\mathrm{d}\mu(x)\neq\intop_{X}f(x)\mathrm{d}\nu(x)$.

Lemma 2.18**.**

Assume that $\mu_{k}$ converges to both $\mu$ and $\nu$ weakly then $\mu=\nu$.

Lemma 2.18 follows from Lemma 2.16.

Lemma 2.19**.**

Let $f,f_{k}\in C_{b}(X)$ ($k\in\mathbb{N}$) such that $\lim_{k\rightarrow\infty}\left\|f-f_{k}\right\|_{\infty}=0$. Assume that $\mu$ and $\nu$ are finite Borel measures on $X$ such that $\intop_{X}f_{k}(x)\mathrm{d}\mu(x)=\intop_{X}f_{k}(x)\mathrm{d}\nu(x)$ for every $k\in\mathbb{N}$. Then $\intop_{X}f(x)\mathrm{d}\mu(x)=\intop_{X}f(x)\mathrm{d}\nu(x)$.

Proof.

Let $\varepsilon>0$ and $k\in\mathbb{N}$ be such that $\left\|f-f_{k}\right\|_{\infty}<\varepsilon$. Then

[TABLE]

[TABLE]

Hence the statement follows. ∎

Lemma 2.20**.**

Let $\Psi\subseteq C_{b}(X)$ be a dense subset with respect to the supremum norm. Assume that $\mu$ and $\nu$ are finite Borel measures on $X$ such that $\intop_{X}f(x)\mathrm{d}\mu(x)=\intop_{X}f(x)\mathrm{d}\nu(x)$ for every $f\in\Psi$. Then $\mu=\nu$.

Proof.

It follows from Lemma 2.19 that $\intop_{X}f(x)\mathrm{d}\mu(x)=\intop_{X}f(x)\mathrm{d}\nu(x)$ for every $f\in C_{b}(X)$. Hence by Lemma 2.16 the statement follows. ∎

Definition 2.21**.**

The Prohorov distance between two finite Borel measures $\mu$ and $\nu$ on $X$ is

[TABLE]

where $\mathcal{B}(X)$ denotes the set of Borel subsets of $X$.

Lemma 2.22**.**

We have that

[TABLE]

Lemma 2.23**.**

We have that

[TABLE]

The statement of Lemma 2.22 and Lemma 2.23 follow from the definition of the Prohorov distance.

Proposition 2.24**.**

The Prohorov distance is a separable metric on the set of all finite Borel measures $\mathcal{M}_{+}(X)$. We have that $\mu_{k}$ converges to $\mu$ weakly if and only if $\lim_{k\rightarrow\infty}\pi(\mu_{k},\mu)=0$. (Note that throughout the paper we assume that $X$ is a separable metric space.) If we further assume that $(X,d)$ is a complete metric space then the Prohorov distance is a complete metric.

For the proof see the section of [2, Theorem 6.8]

Lemma 2.25**.**

If $K\subseteq X$ is compact then $C_{b}(K)$ is separable.

See [7, page 437].

Lemma 2.26**.**

Let $K\subseteq X$ be a compact subset, let $\Psi\subseteq C_{b}(K)$ be a dense subset with respect to the supremum norm and let $\mu_{k}$ be a sequence of finite, Borel measures on $K$ such that $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to a limit $S(f)<\infty$ for every $f\in\Psi$. Then $\mu_{k}(K)$ is bounded and $\mu_{k}$ converges weakly to a finite, Borel measure.

Proof.

Let $g\in\Psi$ such that $\left\|\chi_{K}-g\right\|<1/2$. Then $\chi_{K}\leq 2g$ on $K$ thus

[TABLE]

and so $\mu_{k}(K)$ is bounded.

Since $\mu_{k}(K)$ is bounded and $K$ is a compact metric space it follows, by [14, (17.22) Theorem], that there exists a subsequence $n_{k}$ of $\mathbb{N}$ such that $\mu_{n_{k}}$ weakly converges to a Borel measure $\tau$ of finite total mass. Then

[TABLE]

for every $g\in\Psi$. Let $f$ be a bounded continuous function and $g\in\Psi$ be such that $\left\|f-g\right\|_{\infty}<\varepsilon$. Then

[TABLE]

[TABLE]

[TABLE]

By taking $\varepsilon$ goes to [math] it follows that $\intop_{X}f(x)\mathrm{d}\tau(x)=\lim_{k\rightarrow\infty}\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$. ∎

Definition 2.27**.**

Let $\mu$ and $\mu_{k}$ ($k\in\mathbb{N}$) be locally finite Borel measures on $X$. We say that $\mu_{k}$* vaguely converges to $\mu$* (as $k$ goes to $\infty$) if

[TABLE]

for every $f\in C_{c}(X)$.

Lemma 2.28**.**

Let $X$ be locally compact. There exists $\Psi\subseteq C_{c}(X)$ countable and dense subset with respect to the supremum norm such that if $\mu_{k}$ is a sequence of locally finite Borel measures on $X$ such that $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to a finite limit $S(f)$ for every $f\in\Psi$ then $\mu_{k}$ vaguely converges to a locally finite Borel measure.

Proof.

Since $X$ is a separable metric space it satisfies the Lindelöf property. Thus, because $X$ is locally compact, we can find a sequence of open sets $G_{1}\subseteq G_{2}\subseteq\dots$ such that $\cup_{i=1}^{\infty}G_{i}=X$ and $\overline{G_{i}}$ is compact for every $i\in\mathbb{N}$. Hence for every compact set $K$ there exists $i\in\mathbb{N}$ such that $K\subseteq G_{i}$. We can further assume that $\overline{G_{i}}\subseteq G_{i+1}$. By Lemma 2.25 we can find a countable and dense subset $\Psi_{i}$ of $\left\{f\in C_{b}(X):\mathrm{supp}(f)\subseteq\overline{G_{i}}\right\}$. Let $\Psi=\cup_{i=1}^{\infty}\Psi_{i}$.

Let $\mu_{k}$ be a sequence of deterministic, locally finite, Borel measures on $X$ such that $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to a finite limit $S(f)$ for every $f\in\Psi$. If $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to a finite limit $S(f)$ for every $f\in C_{c}(X)$ then clearly $S$ is a positive linear functional, hence by the Riesz-Markov theorem [20, Theorem 2.14] there exists a locally finite, Borel measures $\mu$ on $X$ such that $S(f)=\intop_{X}f(x)\mathrm{d}\mu(x)$ for every $f\in C_{c}(X)$. Hence the statement would follow. Thus to finish the proof we need to show that $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to a finite limit $S(f)$ for every $f\in C_{c}(X)$.

Let $f\in C_{c}(X)$, let $K=\mathrm{supp}(f)$. There exists $i\in\mathbb{N}$ such that $K\subseteq G_{i}\subseteq\overline{G_{i}}\subseteq G_{i+1}$. By Tietze‘s extension theorem we can find $h_{0}\in C_{c}(X)$ such that $h_{0}(x)=1$ for $x\in\overline{G_{i}}$ and $h_{0}(x)=0$ for $x\notin G_{i+1}$. There exists $h\in\Psi$ such that $\left\|h_{0}-h\right\|_{\infty}<1/2$. Then $\limsup_{k\rightarrow\infty}\mu_{k}(\overline{G_{i}})\leq 2\lim_{k\rightarrow\infty}\intop_{X}h(x)\mathrm{d}\mu_{k}(x)=2S(h)$.

Let $g_{n}\in\Psi_{i}$ be such that $\left\|f-g_{n}\right\|_{\infty}<1/n$. Then

[TABLE]

[TABLE]

Thus for every $n\in\mathbb{N}$ there exists $N\in\mathbb{N}$ such that $\left|\intop_{X}f(x)\mathrm{d}\mu_{k}(x)-S(g_{n})\right|\leq(2S(h)+1)/n$ for every $k\geq N$ and so $\left|\intop_{X}f(x)\mathrm{d}\mu_{k}(x)-\intop_{X}f(x)\mathrm{d}\mu_{j}(x)\right|\leq(4S(h)+2)/n$ for every $j,k\geq N$. Hence $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ is a Cauchy sequence so it has a finite limit $S(f)$ in $\mathbb{R}$. ∎

Lemma 2.29**.**

Let $X$ be locally compact, let $\nu$ and $\mu$ be locally finite Borel measures on $X$ and let $\Psi\subseteq C_{c}(X)$ be as in Lemma 2.28. If $\intop_{X}f(x)\mathrm{d}\nu(x)=\intop_{X}f(x)\mathrm{d}\mu(x)$ for every $f\in\Psi$ then $\mu=\nu$.

Proof.

Let $f\in C_{c}(X)$ be fixed. Similarly to the proof of Lemma 2.28 it can be shown that there exist $h\in\Psi$ and $g_{n}\in\Psi$ for every $n\in\mathbb{N}$ such that

[TABLE]

[TABLE]

Thus $\intop_{X}f(x)\mathrm{d}\mu(x)=\intop_{X}f(x)\mathrm{d}\nu(x)$ for every $f\in C_{c}(X)$ and the statement of the lemma follows by Lemma 2.17. ∎

Proposition 2.30**.**

Assume that $X$ is locally compact. The set of all locally finite Borel measures $\mathcal{M}_{l}(X)$ can be equipped with a complete separable metric such that $\mu_{k}$ converges to $\mu$ in the induced topology if and only if $\mu_{k}$ converges to $\mu$ vaguely.

Proposition 2.30 is shown for $X=\mathbb{R}^{d}$ in [17, Chapter 14] in the section called ‘A metric on measures’ starting on page 194. The proof goes similarly for locally compact separable metric spaces, by replacing the role of balls $B(0,n)$ by an increasing sequence of compact sets $K_{n}$ such that $\cup_{n\in\mathbb{N}}K_{n}=X$ (which we can find since $X$ satisfies the Lindelöf property and is locally compact). We omit the details of the proof.

Lemma 2.31**.**

Let $\nu$ and $\tau$ be two locally finite Borel measures on $X$ such that $\tau(G)\leq\nu(G)$ for every open set $G$. Then $\tau(A)\leq\nu(A)$ for every Borel set $A$.

Lemma 2.31 follows from the outer regularity of the measures.

Lemma 2.32**.**

Let $D\subseteq X$ be a countable and dense subset and let

[TABLE]

Let $\nu$ and $\tau$ be two locally finite Borel measures on $X$ such that $\tau(G)=\nu(G)$ for every $G\in\mathcal{G}$. Then $\tau(A)=\nu(A)$ for every Borel set $A$.

Proof.

Let $A\subseteq X$ be an open set. Then

[TABLE]

is a countable open cover of $A$, which we can enumerate $\left\{B(x_{i},r_{i})\right\}_{i=1}^{\infty}$, thus $A=\cup_{i=1}^{\infty}B(x_{i},r_{i})$. Then

[TABLE]

We can conclude that $\tau(A)=\nu(A)$ for every open set $A\subseteq X$ and so for every Borel set $A\subseteq X$ by Lemma 2.31. ∎

Proposition 2.33**.**

Let $(\Omega,\mathcal{B})$ be a measurable space, let $\nu$ and $\tau$ be $\mathcal{B}$-measurable finite measures on $\Omega$. Let $\mathcal{S}$ be a semiring of sets of $\mathcal{B}$ that generates the $\sigma$-algebra $\mathcal{B}$, assume that $\Omega\in\mathcal{S}$ and $\nu(S)\leq\tau(S)$ for every $S\in\mathcal{S}$. Then $\nu(A)\leq\tau(A)$ for every $A\in\mathcal{B}$.

Proof.

By [16, Section 1.5.1] we have that $\nu$ and $\tau$ are uniquely determined by their values on $\mathcal{S}$, and $\nu$ and $\tau$ equal to their Charatéodry extension from $\mathcal{S}$. Hence $\nu(A)\leq\tau(A)$ for every $A\in\mathcal{B}$ by the definition of Charatéodry extension [16, Section 1.4.4]. ∎

3 Convergence of random measures

We combine the convergence of random variables in probability and the convergence of measures to obtain the convergence of measures in probability. This section includes four subsections. Section 3.1 develops the theory of weak convergence of random measures subsequentially in probability. In Section 3.2 we briefly introduce the concept of weak convergence of random measures in probability. In Section 3.3 we discuss the vague convergence of random measures in probability in the situation when $X$ is a locally compact space. Finally, in Section 3.4 we use the results on the convergence of random measures to obtain some general results about the conditional measure of deterministic measures on random sets, including Theorem 1.16 and Proposition 1.14.

Definition 3.1**.**

The set of all finite Borel measures $\mathcal{M}_{+}(X)$ on $X$ equipped with the weak*∗*-topology on the dual space of $C_{b}(X)$ is a separable metrizable topological space (see Proposition 2.24). A random, finite, Borel measure is an element of $\mathcal{L}^{0}\left(\mathcal{M}_{+}(X)\right)$, i.e. a finite Borel measure valued random variable.

Lemma 3.2**.**

Let $\mu_{k}$ be a sequence of random, finite Borel measures. If there exists $H\in\mathcal{A}$ with $P(H)=1$ such that for every outcome $\omega\in H$ we have that $\mu_{k}$ weakly converges to a finite, Borel measure $\mu$ (note that $\mu$ depends on $\omega\in H$) then $\mu$ is a random, finite Borel measure.

The lemma follows from Lemma 2.7 and basically states that the pointwise limit of random measures is a random measure. We will use this fact throughout the paper without referring to it.

3.1 Weak convergence subsequentially in probability

Definition 3.3**.**

Let $\mu$ and $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$. We say that $\mu_{k}$* weakly converges to $\mu$ subsequentially in probability* if for every subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ there exists a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ and an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\mu_{\beta_{k}}$ converges weakly to $\mu$ on the event $H$.

Remark 3.4*.*

It follows from the Definition 3.21 that if $\mu_{k}$ is a sequence of random, finite, Borel measures on $X$ such that $\mu_{k}$ weakly converges to a random, finite, Borel measure $\mu$ almost surely then $\mu_{k}$ weakly converges to $\mu$ subsequentially in probability.

Proposition 3.5**.**

The limit in Definition 3.3 is unique in $\mathcal{L}^{0}\left(\mathcal{M}_{+}(X)\right)$ if exists.

Proof.

Assume that a sequence of random, finite Borel measures converges weakly to both of the random, finite Borel measures $\mu$ and $\nu$ subsequentially in probability. Then there exists $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ and an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\mu_{\alpha_{k}}$ weakly converges to both $\mu$ and $\nu$ on the event $H$. Hence $\mu=\nu$ on the event $H$ by Lemma 2.18. ∎

Definition 3.6**.**

Let $\mu$ and $\nu$ be two random, finite, Borel measures. We define

[TABLE]

where $\pi$ is the Prohorov distance defined in Definition 2.21.

The following proposition shows that the weak convergence subsequentially in probability is the same as the convergence in probability of Borel random variables that take values in the metric space $(\mathcal{M}_{+}(X),\pi)$. We use this equivalence, of the weak convergence subsequentially in probability and the convergence of $\rho_{\pi}$, throughout the paper without referencing to it.

Proposition 3.7**.**

We have that $\rho_{\pi}$ is a metric on $\mathcal{L}^{0}\left(\mathcal{M}_{+}(X)\right)$ and $\mu_{k}$ weakly converges to $\mu$ subsequentially in probability if and only if $\lim_{k\rightarrow\infty}\rho_{\pi}(\mu_{k},\mu)=0$.

Proof.

The fact that $\rho_{\pi}$ is a metric can be shown similarly to that $\rho$ is a metric, depending on the fact that $x/(1+x)$ is monotone increasing concave function on the positive reals, for reference see [5, Theorem 3.5]. The statement follows from Proposition 2.24 and Remark 2.4. ∎

Proposition 3.8**.**

Let $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$. If $\mu_{k}$ weakly converges to a random, finite, Borel measure $\mu$ subsequentially in probability then $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to $\intop_{X}f(x)\mathrm{d}\mu(x)$ in probability for every $f\in C_{b}(X)$.

Proof.

Let $f\in C_{b}(X)$. For every subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ there exists a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ and an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\mu_{\beta_{k}}$ converges weakly to $\mu$ on the event $H$. Thus $\intop_{X}f(x)\mathrm{d}\mu_{\beta_{k}}(x)$ converges to $\intop_{X}f(x)\mathrm{d}\mu(x)$ almost surely. Hence $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to $\intop_{X}f(x)\mathrm{d}\mu(x)$ in probability by Remark 2.4. ∎

Proposition 3.9**.**

Let $K\subseteq X$ be a compact subset, let $\Psi\subseteq C_{b}(K)$ be a countable dense subset with respect to the supremum norm and let $\mu$ and $\mu_{k}$ be a sequence of random, finite Borel measures on $K$. If $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to $\intop_{X}f(x)\mathrm{d}\mu(x)$ in probability for every $f\in\Psi$ then $\mu_{k}$ weakly converges to $\mu$ subsequentially in probability.

Proof.

By Lemma 2.8 for every subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ there exists a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ and there exists an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\intop_{X}f(x)\mathrm{d}\mu_{\beta_{k}}(x)$ converges to $\intop_{X}f(x)\mathrm{d}\mu(x)$ for every $f\in\Psi$ on the event $H$. We have that $\mu_{\beta_{k}}$ weakly converges to a measure $\tau$ on the event $H$ by Lemma 2.26. We have that $\intop_{X}f(x)\mathrm{d}\mu(x)=\intop_{X}f(x)\mathrm{d}\tau(x)$ for every $f\in\Psi$ on the event $H$ and hence $\tau=\mu$ on the event $H$ by Lemma 2.20. So $\mu_{\beta_{k}}$ weakly converges to the measure $\mu$ on the event $H$. ∎

Proposition 3.10**.**

Let $K\subseteq X$ be a compact subset, let $\Psi\subseteq C_{b}(K)$ be a countable dense subset with respect to the supremum norm and let $\mu_{k}$ be a sequence of random Borel measures on $K$ such that $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges in probability to a random limit $S(f)$ for every $f\in\Psi$ and $\left|S(f)\right|<\infty$ almost surely. Then $\mu_{k}$ weakly converges to a random, finite, Borel measure $\mu$ subsequentially in probability.

Proof.

By Lemma 2.8 for every subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ there exists a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ and there exists an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\intop_{X}f(x)\mathrm{d}\mu_{\beta_{k}}(x)$ converges to $S(f)$ for every $f\in\Psi$ on the event $H$. Then $\mu_{\beta_{k}}$ weakly converges to a random, finite, Borel measure $\mu$ on the event $H$ by Lemma 2.26 and Lemma 3.2. Thus $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to $S(f)=\intop_{X}f(x)\mathrm{d}\mu(x)$ in probability for every $f\in\Psi$ and so $\mu_{k}$ weakly converges to $\mu$ subsequentially in probability by Proposition 3.9. ∎

Proposition 3.11**.**

Let $\nu$ be a deterministic Borel measure on $X$ and $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$ such that $\mu_{k}\ll\nu$ almost surely for every $k$, there exists $c>0$ such that $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c$ for every $k\in\mathbb{N}$, $x\in X$ and $\mu_{k}(A)$ converges in probability for every compact set $A\subseteq X$. Let $f:X\longrightarrow\mathbb{R}$ be a Borel measurable function such that $\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)<\infty$. Then $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to a random variable $Y$ in probability and $E(\left|Y\right|)\leq c\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)$.

Proof.

It is enough to prove the statement for a nonnegative $f$. We have that

[TABLE]

for every nonnegative Borel function $g$ by Fubini‘s theorem. Let $g_{n}(x)=\sum_{i=1}^{N_{n}}b_{i,n}\cdot\chi_{A_{i,n}}(x)$, where $0\leq b_{i,n}<\infty$, $N_{n}\in\mathbb{N}$ and $A_{i,n}\subseteq X$ are compact subsets, such that $g_{n}\leq f$ on $X$ and $0\leq\intop_{X}f(x)-g_{n}(x)\mathrm{d}\nu(x)<1/n$, note that by the definition of Lebesgue integration we can find such $g_{n}$ with $A_{i,n}$ being Borel sets and by inner regularity we can further assume the $A_{i,n}$ to be compact. By assumption $\intop_{X}g_{n}(x)\mathrm{d}\mu_{k}(x)$ converges in probability as $k$ goes to infinity to a random variable $Y_{n}$, thus $\lim_{k\rightarrow\infty}\rho\left(\intop_{X}g_{n}(x)\mathrm{d}\mu_{k}(x),Y_{n}\right)=0$. Hence

[TABLE]

[TABLE]

[TABLE]

and so there exists $m_{n}\in\mathbb{N}$ such that $\rho\left(\intop_{X}f(x)\mathrm{d}\mu_{k}(x),Y_{n}\right)<c/n$ for every $k\geq m_{n}$. If $k\geq\max\left\{m_{n},m_{l}\right\}$ then

[TABLE]

It follows that $Y_{n}$ is a Cauchy sequence for the metric $\rho$, which is a complete metric. Let $Y$ be the limit of $Y_{n}$ in probability. Then

[TABLE]

By taking limit $n$ goes to infinity it follows that $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to $Y$ in probability. By applying Lemma 2.11 twice and by (3.1)

[TABLE]

∎

Corollary 3.12**.**

Let $\nu$ be a deterministic Borel measure on $X$ such that $\mathrm{supp}\nu$ is compact. Let $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$ such that $\mu_{k}\ll\nu$ almost surely for every $k$, there exists $c>0$ such that $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c$ for every $k\in\mathbb{N}$, $x\in X$ and $\mu_{k}(A)$ converges to a random variable $\mu_{\infty}(A)$ in probability for every compact set $A\subseteq X$. Then $\mu_{k}$ weakly converges to a random, finite, Borel measure $\mu$ subsequentially in probability.

Proof.

There exists $\Psi\subseteq C_{b}(\mathrm{supp}\nu)$ countable and dense subset by Lemma 2.25. The conditions of Proposition 3.10 are satisfied for $K=\mathrm{supp}\nu$ by Proposition 3.11. Thus there exists a random Borel measure $\mu$ such that $\mu_{k}$ weakly converges to $\mu$ subsequentially in probability. ∎

Lemma 3.13**.**

Let $g:X\longrightarrow\mathbb{R}$ be a nonnegative Borel function and let $\nu$ be a deterministic finite Borel measure on $X$. Let $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$ such that $\mu_{k}\ll\nu$ almost surely for every $k$ and there exists $c>0$ such that $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c$ for every $k\in\mathbb{N}$, $x\in X$. If $\mu_{k}$ weakly converges to a random, finite, Borel measure $\mu$ subsequentially in probability then $E\left(\intop_{X}g(x)\mathrm{d}\mu(x)\right)\leq c\cdot\intop_{X}g(x)\mathrm{d}\nu(x)$ and $E\left(\intop_{X}g(x)\mathrm{d}\mu_{k}(x)\right)\leq c\cdot\intop_{X}g(x)\mathrm{d}\nu(x)$.

Proof.

Let $G\subseteq X$ be an open set. Then

[TABLE]

by Fubini‘s theorem. Since it holds for every open set $G$ it follows by Lemma 2.31 that

[TABLE]

for every Borel set $A$. Hence

[TABLE]

Let again $G\subseteq X$ be an open set. There exists a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ and an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\mu_{\beta_{k}}$ weakly converges to $\mu$ on the event $H$. Then by Remark 2.15, Fatou‘s lemma and (3.2)

[TABLE]

[TABLE]

Since it holds for every open set $G$ it follows by Lemma 2.31 that $E\left(\mu(A)\right)\leq c\cdot\nu(A)$ for every Borel set $A$ and hence

[TABLE]

∎

Proposition 3.14**.**

Let $\mu^{i}$ and $\mu_{k}^{i}$ be a sequence of random, finite, Borel measures on $X$ for every $i\in\mathbb{N}$. Assume that $\mu_{k}^{i}$ weakly converges to $\mu^{i}$ subsequentially in probability for every $i\in\mathbb{N}$ as $k$ goes to $\infty$. Assume that for every $\varepsilon>0$ there exist $N,n_{0}\in\mathbb{N}$ such that $\sum_{i=N}^{\infty}E(\mu_{k}^{i}(X))<\varepsilon$ for every $k\geq n_{0}$. Then there exists $n_{1}\in\mathbb{N}$ such that $\sum_{i\in\mathbb{N}}\mu_{k}^{i}$ is a sequence of random, finite Borel measures, for $k\geq n_{1}$, that weakly converges to the random, finite Borel measure $\sum_{i\in\mathbb{N}}\mu^{i}$ subsequentially in probability.

Proof.

Since there exists $N_{1},n_{1}\in\mathbb{N}$ such that $\sum_{i=N_{1}}^{\infty}E(\mu_{k}^{i}(X))\leq 1$ for every $k\geq n_{1}$ it follows that $\sum_{i=N_{1}}^{\infty}\mu_{k}^{i}(X)<\infty$ almost surely for $k\geq n_{1}$ and so $\sum_{i=1}^{\infty}\mu_{k}^{i}(X)<\infty$ almost surely for $k\geq n_{1}$ . Thus $\sum_{i=1}^{\infty}\mu_{k}^{i}$ is a random, finite, Borel measure.

Since $\mu_{k}^{i}$ weakly converges to $\mu^{i}$ subsequentially in probability it follows by Proposition 3.8 that $\mu_{k}^{i}(X)$ converges to $\mu^{i}(X)$ in probability as $k$ goes to $\infty$. Thus we can find, by Lemma 2.8, a subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ and an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\mu_{\alpha_{k}}^{i}(X)$ converges to $\mu^{i}(X)$ on $H$ as $k$ goes to $\infty$ for every $i\in\mathbb{N}$. Then by Fatou‘s lemma and Fubini‘s theorem

[TABLE]

Thus $\sum_{i=N_{1}}^{\infty}\mu^{i}(X)<\infty$ almost surely and so $\sum_{i=1}^{\infty}\mu^{i}(X)<\infty$ almost surely. Hence $\sum_{i=1}^{\infty}\mu^{i}$ is a random, finite, Borel measure.

Let $\varepsilon>0$ be fixed and let $N,n_{0}\in\mathbb{N}$ such that $\sum_{i=N}^{\infty}E(\mu_{k}^{i}(X))<\varepsilon$ for every $k\geq n_{0}$. Similarly to (3.3) we have that $E(\sum_{i=N}^{\infty}\mu^{i}(X))\leq\varepsilon$. Thus

[TABLE]

[TABLE]

[TABLE]

where we used the fact that $\rho_{\pi}(\mu,\nu)\leq E(\pi(\mu,\nu))$ and Lemma 2.23. Thus

[TABLE]

since $\sum_{i=1}^{N-1}\mu_{k}^{i}$ weakly converges to $\sum_{i=1}^{N-1}\mu^{i}$ subsequentially in probability. By taking limit $\varepsilon$ goes to [math] it follows by Proposition 3.7 that $\sum_{i=1}^{\infty}\mu_{k}^{i}$ weakly converges to $\sum_{i=1}^{\infty}\mu^{i}$ subsequentially in probability. ∎

Theorem 3.15**.**

Let $\nu$ be a deterministic, finite, Borel measure on $X$. Let $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$ such that $\mu_{k}\ll\nu$ almost surely for every $k$, there exists $c>0$ such that $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c$ for every $k\in\mathbb{N}$, $x\in X$ and $\mu_{k}(A)$ converges to a random variable $\mu_{\infty}(A)$ in probability for every compact set $A\subseteq X$. Then $\mu_{k}$ weakly converges to a random, finite, Borel measure $\mu$ subsequentially in probability. Furthermore, $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to a random variable $S(f)$ in probability with $E(\left|S(f)\right|)\leq c\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)$ for every $f:X\longrightarrow\mathbb{R}$ Borel measurable function such that $\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)<\infty$. For every countable collection of Borel measurable functions $f_{n}:X\longrightarrow\mathbb{R}$ such that $\intop_{X}\left|f_{n}(x)\right|\mathrm{d}\nu(x)<\infty$ we have that $\intop_{X}f_{n}(x)\mathrm{d}\mu(x)=S(f_{n})$ for every $n$ almost surely.

Proof.

Let $K_{1},K_{2},\dots$ be a sequence of disjoint compact subsets of $X$ as in Lemma 2.12. Let $\nu^{i}=\nu|_{K_{i}}$ and $\mu_{k}^{i}=\mu_{k}|_{K_{i}}$. Then $\mu_{k}=\sum_{i=1}^{\infty}\mu_{k}^{i}$ and $\mu_{k}^{i}$ weakly converges to a random, finite Borel measure $\mu^{i}$ subsequentially in probability by Corollary 3.12. Since $\sum_{i=N}^{\infty}E\left(\mu_{k}^{i}(X)\right)\leq c\sum_{i=N}^{\infty}\nu(K_{i})<\infty$ it follows that the conditions of Proposition 3.14 are satisfied and so $\mu=\sum_{i=1}^{\infty}\mu^{i}$ is a random, finite, Borel measure and $\mu_{k}$ weakly converges to $\mu$ subsequentially in probability.

By Proposition 3.11 it follows that $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to a random variable $S(f)$ in probability with $E(\left|S(f)\right|)\leq c\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)$ for every $f:X\longrightarrow\mathbb{R}$ Borel measurable function such that $\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)<\infty$.

Let $n\in\mathbb{N}$ be fixed. For every $\varepsilon>0$ we can find $g\in C_{b}(X)$ such that $\intop_{X}\left|f_{n}(x)-g(x)\right|\mathrm{d}\nu(x)<\varepsilon$ since $C_{b}(X)$ is dense in $\mathcal{L}^{1}(X)$ (see [15, Proposition 1.3.22]). Then

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where we used Lemma 3.13 and Remark 2.6. Hence taking limit $k$ goes to infinity it follows that

[TABLE]

since $\intop_{X}g(x)\mathrm{d}\mu_{k}(x)$ converges to $\intop_{X}g(x)\mathrm{d}\mu(x)$ in probability by Proposition 3.8 and $\intop_{X}f_{n}(x)\mathrm{d}\mu_{k}(x)$ converges to $S(f_{n})$ in probability by the definition of $S(f_{n})$. Taking limit $\varepsilon$ goes to [math] it follows that $\intop_{X}f_{n}(x)\mathrm{d}\mu(x)=S(f_{n})$ almost surely. ∎

Remark 3.16*.*

In particular, if we take $f_{n}=\chi_{A_{n}}$ in Theorem 3.15 for a countable collection of Borel sets $A_{n}$ then it follows that $\mu(A_{n})=S(\chi_{A_{n}})$ almost surely for every $n$ where $\mu_{k}(A_{n})$ converges to $S(\chi_{A_{n}})$ in probability as $k$ goes to infinity.

Remark 3.17*.*

In Theorem 3.15 we can relax the condition $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c$ for every $k\in\mathbb{N}$, $x\in X$ . It is enough to assume that there exists a nonnegative Borel function $c:X\longrightarrow\mathbb{R}$ such that $\int c(x)\mathrm{d}\nu(x)<\infty$ and $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c(x)$ for $\nu$ almost every $x\in X$ for every $k$. It can be seen by replacing $\mathrm{d}\nu(x)$ by $c(x)\mathrm{d}\nu(x)$.

Proposition 3.18**.**

Let $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$ such that there exists a sequence of random closed sets $F_{1}\supseteq F_{2}\supseteq\dots$ such that $\mathrm{supp}\mu_{k}\subseteq F_{k}$ almost surely. If $\mu_{k}$ weakly converges to a random finite Borel measure $\mu$ subsequentially in probability then $\mathrm{supp}\mu\subseteq\bigcap_{n=1}^{\infty}F_{n}$ almost surely.

Proof.

Let $G_{n}=X\setminus F_{n}$ be a random open set. Let $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ be a subsequence of $\mathbb{N}$ such that $\mu_{\alpha_{k}}$ weakly converges to $\mu$ on the event $H\in\mathcal{A}$ and $P(H)=1$. We can further assume that $\mathrm{supp}\mu_{k}\subseteq F_{k}$ on the event $H$. Then $\mu(G_{n})\leq\liminf_{k\rightarrow\infty}\mu_{\alpha_{k}}(G_{n})=0$ on the event $H$ by Remark 2.15. Hence $\mu\left(\bigcup_{n=1}^{\infty}G_{n}\right)=0$ on the event $H$ and so $\mathrm{supp}\mu\subseteq\bigcap_{n=1}^{\infty}F_{n}$ almost surely. ∎

Lemma 3.19**.**

Let $\nu$ be a deterministic, finite, Borel measure on $X$ with compact support. Let $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$ such that $\mu_{k}\ll\nu$ almost surely for every $k$, there exists $c>0$ such that $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c$ for every $k\in\mathbb{N}$, $x\in X$ and $\mu_{k}(A)$ converges to a random variable $\mu_{\infty}(A)$ in probability for every compact set $A\subseteq X$. Let $f:X\longrightarrow\mathbb{R}$ be a nonnegative Borel measurable function such that $\intop_{X}f(x)\mathrm{d}\nu(x)<\infty$. Then the sequence of random, finite, Borel measures $f(x)\mathrm{d}\mu_{k}(x)$ weakly converges to the random, finite, Borel measure $f(x)\mathrm{d}\mu(x)$ subsequentially in probability where $\mu$ is the random, finite, Borel measure such that $\mu_{k}$ weakly converges to $\mu$ subsequentially in probability.

Proof.

We have that $E(\int_{X}f(x)\mathrm{d}\mu_{k}(x))<c\intop_{X}f(x)\mathrm{d}\nu(x)<\infty$ thus $f(x)\mathrm{d}\mu_{k}(x)$ is a random finite Borel measure almost surely. We can find a countable and dense $\Psi\subseteq C_{b}(\mathrm{supp}(\nu))$ by Lemma 2.25. By Theorem 3.15 it follows that $\intop_{X}g(x)f(x)\mathrm{d}\mu_{k}(x)$ converges to $\intop_{X}g(x)f(x)\mathrm{d}\mu(x)$ in probability for every $g\in\Psi$ and $E(\int_{X}f(x)\mathrm{d}\mu(x))<c\intop_{X}f(x)\mathrm{d}\nu(x)<\infty$ thus $f(x)\mathrm{d}\mu(x)$ is a random finite Borel measure almost surely. It follows that $f(x)\mathrm{d}\mu_{k}(x)$ weakly converges to $f(x)\mathrm{d}\mu(x)$ subsequentially in probability by Proposition 3.9. ∎

Proposition 3.20**.**

Let $\nu$ be a deterministic, finite, Borel measure on $X$. Let $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$ such that $\mu_{k}\ll\nu$ almost surely for every $k$, there exists $c>0$ such that $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c$ for every $k\in\mathbb{N}$, $x\in X$ and $\mu_{k}(A)$ converges to a random variable $\mu_{\infty}(A)$ in probability for every compact set $A\subseteq X$. Let $f:X\longrightarrow\mathbb{R}$ be a nonnegative Borel measurable function such that $\intop_{X}f(x)\mathrm{d}\nu(x)<\infty$. Then the sequence of random, finite, Borel measures $f(x)\mathrm{d}\mu_{k}(x)$ weakly converges to the random, finite, Borel measure $f(x)\mathrm{d}\mu(x)$ subsequentially in probability where $\mu$ is the random, finite, Borel measure such that $\mu_{k}$ weakly converges to $\mu$ subsequentially in probability. Additionally, $\int_{A}f(x)\mathrm{d}\mu_{k}(x)$ converges to $\int_{A}f(x)\mathrm{d}\mu(x)$ in probability for every Borel set $A\subseteq X$.

Proof.

Let $K_{1},K_{2},\dots$ be compact sets, $\nu^{i}=\nu|_{K_{i}}$, $\mu_{k}^{i}=\mu_{k}|_{K_{i}}$, $\mu^{i}$ and $\mu$ as in the proof of Theorem 3.15. Then by Lemma 3.19 it follows that the sequence of measures $f(x)\mathrm{d}\mu_{k}^{i}(x)$ weakly converges to $f(x)\mathrm{d}\mu^{i}(x)$ subsequentially in probability. Since

[TABLE]

it follows by the application of Proposition 3.14 that $f(x)\mathrm{d}\mu(x)=\sum_{i=1}^{\infty}f(x)\mathrm{d}\mu^{i}(x)$ is a random, finite, Borel measure and $f(x)\mathrm{d}\mu_{k}(x)$ weakly converges to $f(x)\mathrm{d}\mu(x)$ subsequentially in probability.

The same argument for $\chi_{A}\cdot f$ in place of $f$ shows that $\chi_{A}(x)f(x)\mathrm{d}\mu_{k}(x)$ weakly converges to $\chi_{A}(x)f(x)\mathrm{d}\mu(x)$ subsequentially in probability, in particular, $\int_{A}f(x)\mathrm{d}\mu_{k}(x)$ converges to $\int_{A}f(x)\mathrm{d}\mu(x)$ in probability by Proposition 3.8. ∎

3.2 Weak convergence in probability

Definition 3.21**.**

Let $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$. We say that $\mu_{k}$* weakly converges to a random, finite, Borel measure $\mu$ in probability* if $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to $\intop_{X}f(x)\mathrm{d}\mu(x)$ in probability for every deterministic, bounded, continuous function $f:X\longrightarrow\mathbb{R}$.

Proposition 3.22**.**

The convergence weakly in probability is induced by the topology which has base elements formed by finite intersection of sets in the form:

[TABLE]

where $f\in C_{b}(X)$, $r>0$ and $Y$ is a real-valued random variable with almost surely finite values.

Proposition 3.22 can be verified easily, we leave the details for the reader.

Proposition 3.23**.**

Let $\mu$ and $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$. If $\mu_{k}$ weakly converges to $\mu$ subsequentially in probability then $\mu_{k}$ weakly converges to $\mu$ in probability.

Proposition 3.23 is a reformulation of Proposition 3.8.

Proposition 3.24**.**

Let $\mu$ and $\mu_{k}$ be a sequence of random, finite, Borel measures supported on a deterministic compact subset $K\subseteq X$. If $\mu_{k}$ weakly converges to $\mu$ in probability then $\mu_{k}$ weakly converges to $\mu$ subsequentially in probability.

Proof.

We can find a countable and dense $\Psi\subseteq C_{b}(K)$ by Lemma 2.25. Hence the statement follows from Proposition 3.9. ∎

Proposition 3.25**.**

Let $\mu_{k}$ be a sequence of random, finite, Borel measures. Assume that $\tau$ and $\nu$ are random, finite, Borel measures on $X$ such that $\mu_{k}(G)$ converges to $\tau(G)$ in probability and $\mu_{k}(G)$ converges to $\nu(G)$ in probability for every open set $G\subseteq X$. Then $\mu=\tau$ almost surely.

Proof.

Let $D\subseteq X$ be a countable and dense subset and let $\mathcal{G}$ be as in Lemma 2.32. Let $G\in\mathcal{G}$. Then $\mu_{k}(G)$ converges to both $\mu(G)$ and $\tau(G)$ in probability hence $\mu(G)=\tau(G)$ almost surely. Since $\mathcal{G}$ is countable it follows that $\mu(G)=\tau(G)$ for every $G\in\mathcal{G}$ almost surely, i.e. there exists an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\mu(G)=\tau(G)$ for every $G\in\mathcal{G}$. Then on the event $H$ we have that $\mu(A)=\tau(A)$ for every Borel set $A\subseteq X$ by Lemma 2.32. Hence $\mu=\tau$ almost surely. ∎

Proposition 3.26**.**

Let $\mu^{i}$ and $\mu_{k}^{i}$ be a sequence of random, finite, Borel measures on $X$ for every $i\in\mathbb{N}$. Assume that $\mu_{k}^{i}$ weakly converges to $\mu^{i}$ in probability for every $i\in\mathbb{N}$ as $k$ goes to $\infty$. Assume that for every $\varepsilon>0$ there exist $N,n_{0}\in\mathbb{N}$ such that $\sum_{i=N}^{\infty}E(\mu_{k}^{i}(X))<\varepsilon$ for every $k\geq n_{0}$. Then there exists $n_{1}\in\mathbb{N}$ such that $\sum_{i\in\mathbb{N}}\mu_{k}^{i}$ is a sequence of random, finite Borel measures, for $k\geq n_{1}$, that weakly converges to the random, finite Borel measure $\sum_{i\in\mathbb{N}}\mu^{i}$ in probability.

The proof of Proposition 3.26 goes similarly to the proof of Proposition 3.14 with the difference that instead of

[TABLE]

we need to estimate

[TABLE]

for a given $f\in C_{b}(X)$. We leave for the reader to check the details. We provide a similar proof in the proof of Proposition 3.40.

3.3 Vague convergence in probability

Throughout this section we assume that $X$ is locally compact.

Definition 3.27**.**

The set of all locally finite Borel measures $\mathcal{M}_{l}(X)$ on $X$ equipped with the weak*∗*-topology on the dual space of $C_{c}(X)$ is a Polish space (see Lemma 2.30). A random, finite, Borel measure is an element of $\mathcal{L}^{0}\left(\mathcal{M}_{l}(X)\right)$, i.e. a locally finite Borel measure valued random variable.

Lemma 3.28**.**

Let $\mu_{k}$ be a sequence of random, locally finite, Borel measures. If there exists $H\in\mathcal{A}$ with $P(H)=1$ such that for every outcome $\omega\in H$ we have that $\mu_{k}$ vaguely converges to a locally finite, Borel measure $\mu$ (note that $\mu$ depends on $\omega\in H$) then $\mu$ is a random, locally finite Borel measure.

The lemma states that the pointwise limit of random locally finite Borel measures is a random locally finite Borel measure and it follows from Lemma 2.7. We use this fact throughout the paper with no reference.

Definition 3.29**.**

Let $\mu$ and $\mu_{k}$ be a sequence of random, locally finite, Borel measures on $X$. We say that $\mu_{k}$* vaguely converges to $\mu$ in probability* if $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to $\intop_{X}f(x)\mathrm{d}\mu(x)$ in probability for every $f\in C_{c}(X)$.

Remark 3.30*.*

It follows from Definition 3.21 that if $\mu_{k}$ is a sequence of random, locally finite, Borel measures on $X$ such that $\mu_{k}$ vaguely converges to a random Borel measure $\mu$ almost surely then $\mu_{k}$ vaguely converges to $\mu$ in probability.

Remark 3.31*.*

It follows from Definition 3.21 that if $\mu_{k}$ is a sequence of random, finite, Borel measures on $X$ such that $\mu_{k}$ weakly converges to a random, finite, Borel measure $\mu$ in probability then $\mu_{k}$ vaguely converges to $\mu$ in probability.

Proposition 3.32**.**

The convergence vaguely in probability is induced by the topology which has base elements formed by finite intersection of sets in the form:

[TABLE]

where $f\in C_{c}(X)$, $r>0$ and $Y$ is a real-valued random variable with almost surely finite values.

Proposition 3.32 can be verified easily, we leave the details for the reader.

Proposition 3.33**.**

Let $X$ be locally compact. The limit in Definition 3.29 is unique in $\mathcal{L}^{0}\left(\mathcal{M}_{l}(X)\right)$ if exists.

Proof.

Let $\mu_{k}$ be a sequence of random, finite, Borel measures on $X$ such that $\mu_{k}$ vaguely converges to a random, finite, Borel measure $\mu$ and also to a random, finite, Borel measure $\tau$ in probability. Let $\Psi\subseteq C_{c}(X)$ as in Lemma 2.28. Then $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to $\intop_{X}f(x)\mathrm{d}\mu(x)$ in probability for every $f\in\Psi$ and also to $\intop_{X}f(x)\mathrm{d}\tau(x)$ in probability. Since $\Psi$ is countable it follows that $\intop_{X}f(x)\mathrm{d}\mu(x)=\intop_{X}f(x)\mathrm{d}\tau(x)$ for every $f\in\Psi$ almost surely. Thus $\mu=\nu$ almost surely by Lemma 2.29. ∎

Lemma 3.34**.**

Let $X$ be locally compact. Let $\mu$ and $\mu_{k}$ be a sequence of random, locally finite, Borel measures on $X$. Assume that for every subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ there exists a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ and there exists an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\mu_{k}$ vaguely converges to $\mu$ on the event $H$ then $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to $\intop_{X}f(x)\mathrm{d}\mu(x)$ in probability for every $f\in C_{c}(X)$.

Lemma 3.34 can be proven similarly to the proof of Proposition 3.8.

Lemma 3.35**.**

Let $X$ be locally compact, let $\Psi\subseteq C_{c}(X)$ as in Lemma 2.28 and $\mu$ and $\mu_{k}$ be a sequence of random, locally finite Borel measures on $X$. If $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to $\intop_{X}f(x)\mathrm{d}\mu(x)$ in probability for every $f\in\Psi$ then for every subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ there exists a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ and there exists an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\mu_{k}$ vaguely converges to $\mu$ on the event $H$.

Lemma 3.35 can be shown similarly to the proof of Proposition 3.9 by replacing the use of Lemma 2.26 by the use of Lemma 2.28 and the use of Lemma 2.20 by Lemma 2.29.

Lemma 3.36**.**

Let $X$ be locally compact, let $\Psi\subseteq C_{c}(X)$ as in Lemma 2.28 and $\mu_{k}$ be a sequence of random, locally finite Borel measures on $X$. Assume that $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges in probability to a random finite limit $S(f)$ for every $f\in\Psi$. Then there exists a random, locally finite Borel measure $\mu$ on $X$ such that for every subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ there exists a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ and there exists an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\mu_{k}$ vaguely converges to $\mu$ on the event $H$.

Lemma 3.36 can be shown similarly to the proof Proposition 3.10 by replacing the use of Lemma 2.26 by the use of Lemma 2.28, the use of Lemma 2.20 by Lemma 2.29 and the use of Proposition 3.9 by the use of Lemma 3.35.

Theorem 3.37**.**

Let $X$ be locally compact and let $\mu$ and $\mu_{k}$ be a sequence of random, locally finite Borel measures on $X$. Then $\mu_{k}$ vaguely converges to $\mu$ in probability if and only if for every subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ there exists a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ and there exists an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\mu_{k}$ vaguely converges to $\mu$ on the event $H$.

Theorem 3.37 follows from Lemma 3.34 and Lemma 3.35.

Remark 3.38*.*

Let $X$ be locally compact. Proposition 3.32 states that the vague convergence in probability is a topological convergence. Due to Theorem 3.37 the ‘convergence vaguely subsequentially in probability’ and the convergence vaguely in probability are the same convergence. Hence it can be shown similarly to Proposition 3.7 that convergence vaguely in probability is also a metrizable convergence using the fact that the vague convergence of deterministic locally finite measures is a metric convergence, see Proposition 2.30. Similarly to Proposition 3.7 the vague convergence in probability is the same as the convergence in probability of Borel random variables that take values in the metric space of $\mathcal{M}_{l}(X)$ equipped with the metric of Proposition 2.30.

Proposition 3.39**.**

Let $X$ be locally compact. Let $\nu$ be a deterministic, locally finite, Borel measure on $X$ and $\mu_{k}$ be a sequence of random, locally finite, Borel measures on $X$ such that $\mu_{k}\ll\nu$ almost surely for every $k$, there exists $c>0$ such that $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c$ for every $k\in\mathbb{N}$, $x\in X$ and $\mu_{k}(A)$ converges in probability for every compact set $A\subseteq X$. Let $f:X\longrightarrow\mathbb{R}$ be a Borel measurable function such that $\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)<\infty$. Then $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to a random variable $Y$ in probability and $E(\left|Y\right|)\leq c\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)$.

The proof of Proposition 3.39 is identical to the proof of Proposition 3.11.

Proposition 3.40**.**

Let $X$ be locally compact. Let $\mu^{i}$ and $\mu_{k}^{i}$ be a sequence of random, locally finite, Borel measures on $X$ for every $i\in\mathbb{N}$. Assume that $\mu_{k}^{i}$ vaguely converges to $\mu^{i}$ in probability for every $i\in\mathbb{N}$ as $k$ goes to $\infty$. Assume that for every compact set $K\subseteq X$ and $\varepsilon>0$ there exist $N\in\mathbb{N}$ such that $\sum_{i=N}^{\infty}E(\mu_{k}^{i}(K))<\varepsilon$ for every $k\in\mathbb{N}$. Then $\sum_{i\in\mathbb{N}}\mu_{k}^{i}$ is a sequence of random, locally finite Borel measures that vaguely converges to the random, locally finite Borel measure $\sum_{i\in\mathbb{N}}\mu^{i}$ in probability.

Proof.

Since for every compact set $K\subseteq X$ there exists $N_{1}\in\mathbb{N}$ such that $\sum_{i=N_{1}}^{\infty}E(\mu_{k}^{i}(K))\leq 1$ for every $k\in\mathbb{N}$ it follows that $\sum_{i=N_{1}}^{\infty}\mu_{k}^{i}(K)<\infty$ almost surely for $k\in\mathbb{N}$ and so $\sum_{i=1}^{\infty}\mu_{k}^{i}(K)<\infty$ almost surely. Thus $\sum_{i=1}^{\infty}\mu_{k}^{i}$ is a random, locally finite, Borel measure since $X$ is locally compact. Let $h\in C_{c}(X)$ such that $h(x)=1$ for every $x\in K$ and $\left\|h\right\|_{\infty}\leq 1$, we can find such $h$ by Tietze‘s extension theorem and the fact that $X$ is locally compact. We can find $N_{2}\in\mathbb{N}$ such that $\sum_{i=N_{2}}^{\infty}E(\mu_{k}^{i}(\mathrm{supp}(h))\leq 1$ for every $k\in\mathbb{N}$. We can find, by Lemma 2.8, a subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ and an event $H\in\mathcal{A}$ with $P(H)=1$ such that $\intop_{X}h(x)\mathrm{d}\mu_{\alpha_{k}}^{i}(x)$ converges to $\intop_{X}h(x)\mathrm{d}\mu^{i}(x)$ on $H$ as $k$ goes to $\infty$ for every $i\in\mathbb{N}$. Then by Fatou‘s lemma and Fubini‘s theorem

[TABLE]

[TABLE]

Thus $\sum_{i=N_{1}}^{\infty}\mu^{i}(K)<\infty$ almost surely and so $\sum_{i=1}^{\infty}\mu^{i}(K)<\infty$ almost surely. Hence $\sum_{i=1}^{\infty}\mu^{i}$ is a random, locally finite, Borel measure.

Let $f\in C_{c}(X)$ and $K=\mathrm{supp}(f)$. Let $\varepsilon>0$ be fixed and let $N\in\mathbb{N}$ such that $\sum_{i=N}^{\infty}E(\mu_{k}^{i}(K))<\varepsilon$ for every $k\in\mathbb{N}$. Similarly to (3.4) we can further assume that $E(\sum_{i=N}^{\infty}\mu^{i}(K))\leq\varepsilon$. Thus

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where we used the fact that $\rho(Y,Z)\leq E(\left|Y-Z\right|)$. Thus

[TABLE]

since $\sum_{i=1}^{N-1}\intop_{X}f(x)\mathrm{d}\mu_{k}^{i}(x))$ converges to $\sum_{i=1}^{N-1}\intop_{X}f(x)\mathrm{d}\mu^{i}(x)$ in probability. By taking limit $\varepsilon$ goes to [math] it follows that $\sum_{i=1}^{\infty}\intop_{X}f(x)\mathrm{d}\mu_{k}^{i}(x)$ converges to $\sum_{i=1}^{\infty}\intop_{X}f(x)\mathrm{d}\mu^{i}(x)$ in probability. ∎

Theorem 3.41**.**

Let $X$ be locally compact. Let $\nu$ be a deterministic, locally finite, Borel measure on $X$. Let $\mu_{k}$ be a sequence of random, locally finite, Borel measures on $X$ such that $\mu_{k}\ll\nu$ almost surely for every $k$, there exists $c>0$ such that $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c$ for every $k\in\mathbb{N}$, $x\in X$ and $\mu_{k}(A)$ converges to a random variable $\mu_{\infty}(A)$ in probability for every compact set $A\subseteq X$. Then $\mu_{k}$ vaguely converges to a random, locally finite, Borel measure $\mu$ in probability. Furthermore, $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to a random variable $S(f)$ in probability with $E(\left|S(f)\right|)\leq c\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)$ for every $f:X\longrightarrow\mathbb{R}$ Borel measurable function such that $\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)<\infty$. For every countable collection of Borel measurable functions $f_{n}:X\longrightarrow\mathbb{R}$ such that $\intop_{X}\left|f_{n}(x)\right|\mathrm{d}\mu(x)<\infty$ we have that $\intop_{X}f_{n}(x)\mathrm{d}\mu(x)=S(f_{n})$ almost surely for every $n$.

Proof.

Due to Remark 2.13 there exists a sequence of disjoint compact sets $A_{i}$ such that $\nu^{i}=\nu|_{A_{i}}$ is a finite measure for every $i\in\mathbb{N}$ and $\nu\left(X\setminus\left(\cup_{i\in\mathbb{N}}A_{i}\right)\right)=0$. Let $\mu_{k}^{i}=\mu_{k}|_{A_{i}}$. Then $\mu_{k}=\sum_{i=1}^{\infty}\mu_{k}^{i}$ and $\mu_{k}^{i}$ vaguely converges to a random, finite Borel measure $\mu^{i}$ in probability by Remark 3.31 and Theorem 3.15. Since $\sum_{i=N}^{\infty}E\left(\mu_{k}^{i}(K)\right)\leq c\sum_{i=N}^{\infty}\nu(K)<\infty$ for every compact set $K$ it follows that the conditions of Proposition 3.40 are satisfied and so $\mu=\sum_{i=1}^{\infty}\mu^{i}$ is a random, locally finite, Borel measure and $\mu_{k}$ vaguely converges to $\mu$ in probability.

By Proposition 3.39 it follows that $\intop_{X}f(x)\mathrm{d}\mu_{k}(x)$ converges to a random variable $S(f)$ in probability with $E(\left|S(f)\right|)\leq c\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)$ for every $f:X\longrightarrow\mathbb{R}$ Borel measurable function such that $\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)<\infty$.

Let $n\in\mathbb{N}$ be fixed. For every $\varepsilon>0$ we can find $g\in C_{c}(X)$ such that $\intop_{X}\left|f_{n}(x)-g(x)\right|\mathrm{d}\nu(x)<\varepsilon$ by [20, Theorem 3.14]. The rest of the proof proceeds similarly to the proof of Theorem 3.15. ∎

Remark 3.42*.*

In particular, if we take $f_{n}=\chi_{A_{n}}$ in Theorem 3.41 for a countable collection of Borel sets $A_{n}$ such that $\nu(A_{n})<\infty$ then it follows that $\mu(A_{n})=S(\chi_{A_{n}})$ almost surely for every $n$ where $\mu_{k}(A_{n})$ converges to $S(\chi_{A_{n}})$ in probability as $k$ goes to infinity.

Remark 3.43*.*

In Theorem 3.41 we can relax the condition $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c$ for every $k$. It is enough to assume that there exists a nonnegative Borel function $c:X\longrightarrow\mathbb{R}$ such that $\intop_{K}c(x)\mathrm{d}\nu(x)<\infty$ for every compact set $K$ and $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c(x)$ for $\nu$ almost every $x\in X$ for every $k$ . It can be seen by replacing $\mathrm{d}\nu(x)$ by $c(x)\mathrm{d}\nu(x)$.

Proposition 3.44**.**

Let $\mu_{k}$ be a sequence of random, locally finite, Borel measures on $X$ such that there exists a sequence of random closed sets $F_{1}\supseteq F_{2}\supseteq\dots$ such that $\mathrm{supp}\mu_{k}\subseteq F_{k}$ almost surely. If $\mu_{k}$ vaguely converges to a random, locally finite, Borel measure $\mu$ in probability then $\mathrm{supp}\mu\subseteq\bigcap_{n=1}^{\infty}F_{n}$ almost surely.

Proposition 3.44 can be shown similarly to the proof Proposition 3.18 due to the equivalence in Theorem 3.37.

Proposition 3.45**.**

Let $X$ be locally compact. Let $\nu$ be a deterministic, locally finite, Borel measure on $X$. Let $\mu_{k}$ be a sequence of random, locally finite, Borel measures on $X$ such that $\mu_{k}\ll\nu$ almost surely for every $k$, there exists $c>0$ such that $E\left(\frac{\mathrm{d}\mu_{k}}{\mathrm{d}\nu}(x)\right)\leq c$ for every $k\in\mathbb{N}$, $x\in X$ and $\mu_{k}(A)$ converges to a random variable $\mu_{\infty}(A)$ in probability for every compact set $A\subseteq X$. Let $f:X\longrightarrow\mathbb{R}$ be a nonnegative Borel measurable function such that for every $y\in X$ there exists a neigbourhood $U$ of $y$ such that $\intop_{U}f(x)\mathrm{d}\nu(x)<\infty$. Then the sequence of random, locally finite, Borel measures $f(x)\mathrm{d}\mu_{k}(x)$ vaguely converges to the random, locally finite, Borel measure $f(x)\mathrm{d}\mu(x)$ in probability where $\mu$ is the random, locally finite, Borel measure such that $\mu_{k}$ weakly converges to $\mu$ subsequentially in probability. Additionally, $\int_{A}f(x)\mathrm{d}\mu_{k}(x)$ converges to $\int_{A}f(x)\mathrm{d}\mu(x)$ in probability for any compact set $A\subseteq X$.

Proof.

Let $U$ be an open set such that $\intop_{U}f(x)\mathrm{d}\nu(x)<\infty$. Then $E(\intop_{U}f(x)\mathrm{d}\mu_{k}(x))\leq c\intop_{U}f(x)\mathrm{d}\nu(x)<\infty$ and by Theorem 3.41 we have that $E(\intop_{U}f(x)\mathrm{d}\mu(x))\leq c\intop_{U}f(x)\mathrm{d}\nu(x)<\infty$. Hence $f(x)\mathrm{d}\mu_{k}(x)$ and $f(x)\mathrm{d}\mu(x)$ are random, locally finite, Borel measures.

Let $\Psi\subseteq C_{c}(X)$ as in Lemma 2.28. By Theorem 3.41 it follows that $\intop_{X}g(x)f(x)\mathrm{d}\mu_{k}(x)$ converges to $\intop_{X}g(x)f(x)\mathrm{d}\mu(x)$ in probability for every $g\in\Psi$.

By the application of Theorem 3.41 to the reference measure $f(x)\mathrm{d}\nu(x)$ and the sequence $f(x)\mathrm{d}\mu_{k}(x)$ it follows that the sequence of random, locally finite, Borel measures $f(x)\mathrm{d}\mu_{k}(x)$ converges to a random, locally finite, Borel measure $\tau$ and $\intop_{X}g(x)f(x)\mathrm{d}\mu_{k}(x)$ converges to $\intop_{X}g(x)\mathrm{d}\tau(x)$ in probability for every $g\in\Psi$.

Thus $\intop_{X}g(x)f(x)\mathrm{d}\mu(x)=\intop_{X}g(x)\mathrm{d}\tau(x)$ for every $g\in\Psi$ almost surely. Hence $f(x)\mathrm{d}\mu(x)=\mathrm{d}\tau(x)$ almost surely by Lemma 2.29.

Applying Theorem 3.41 to the function $\chi_{A}\cdot f$ for a compact set $A\subseteq X$, it follows that $\int_{A}f(x)\mathrm{d}\mu_{k}(x)$ converges to $\int_{A}f(x)\mathrm{d}\mu(x)$ in probability. ∎

3.4 General existence of the conditional measure

In this section our main goal is to prove Theorem 1.16 and Proposition 1.14. The proof of Proposition 1.14 can be found at the end of the section and Theorem 1.16 follows from Theorem 3.49, Theorem 3.53 and Lemma 3.50.

Lemma 3.46**.**

Let $A\subseteq X$ be a Borel set $\nu=\sum_{i=1}^{\infty}\nu^{i}$ be a finite Borel measure and $\mu_{k}^{i}$ be random finite Borel measures for every $i\in\mathbb{N}$ such that $\mu_{k}^{i}(A)$ converges in $\mathcal{L}^{1}$ to a random variable $\mu^{i}(A)$ for every $i\in\mathbb{N}$ and $E(\mu_{k}^{i}(A))=\nu^{i}(A)$. Then $\sum_{i=1}^{\infty}\mu_{k}^{i}(A)$ converges in $\mathcal{L}^{1}$ to $\sum_{i=1}^{\infty}\mu^{i}(A)$ as $k$ goes to $\infty$.

Proof.

Since $\mu_{k}^{i}(A)$ converges to $\mu^{i}(A)$ in $\mathcal{L}^{1}$ it follows that $E(\mu^{i}(A))=E(\mu_{k}^{i}(A))=\nu^{i}(A)$. Thus by Fubini‘s theorem that

[TABLE]

for every $n\in\mathbb{N}$ and $\sum_{i=1}^{\infty}\nu^{i}(A)=\nu(A)<\infty$. Let $\eta>0$ and let $n\in\mathbb{N}$ be large enough that $\sum_{i=n}^{\infty}\nu^{i}(A)<\eta$. By assumption $\sum_{i=1}^{n}\mu_{k}^{i}(A)$ converges in $\mathcal{L}^{1}$ to $\sum_{i=1}^{n}\mu^{i}(A)$, hence converges in probability. Thus

[TABLE]

[TABLE]

[TABLE]

Hence by taking limit $\eta$ goes to [math] it follows that $\sum_{i=1}^{\infty}\mu_{k}^{i}(A)$ converges in probability to $\sum_{i=1}^{\infty}\mu^{i}(A)$. Thus the statement follows from Lemma 2.10. ∎

Lemma 3.47**.**

Let $\nu$ be a finite Borel measure on $X$ such that $\nu(X\setminus X_{0})=0$ and let $A\subseteq X$ be a Borel set. Assume that $\mathcal{C}_{k}(\nu|_{D})(X)=\mathcal{C}_{k}(\nu)(D)$ converges in $\mathcal{L}^{1}$ for every compact set $D\subseteq X_{0}$ with $I_{\varphi}(\nu|_{D})<\infty$. Assume that $\mathcal{C}_{k}(\nu_{\perp})(X)$ converges to [math] in probability. Then $\mathcal{C}_{k}(\nu_{R})(A)$ converges in $\mathcal{L}^{1}$ to a limit $\mu_{\infty}(A)$, and both $\mathcal{C}_{k}(\nu)(A)$ and $\mathcal{C}_{k}(\nu_{R})(A)$ converges in probability to the same limit $\mu_{\infty}(A)$ and $E(\mu_{\infty}(A))=\nu_{R}(A)\leq\nu(A)$.

Proof.

Let us take a sequence $(A_{n})_{n\in\mathbb{N}}$ as in Proposition 1.7 for the measure $\nu|_{A}$ in place of $\nu$. Note that we can assume that $A_{n}\subseteq A\cap X_{0}$ for every $n\in\mathbb{N}$ because $\nu|_{A}(X\setminus X_{0})=0$. By Lemma 2.12 we can further assume that all the $A_{n}$ are compact. Then $\nu|_{A}=\nu_{\bot}|_{A}+\sum_{n\in\mathbb{N}}\nu|_{A_{n}}$ and $I_{\varphi}(\nu|_{A_{n}})<\infty$. By the assumption of the statement it follows that $\mathcal{C}_{k}(\nu|_{A_{n}})(X)=\mathcal{C}_{k}(\nu)(A_{n})$ converges in $\mathcal{L}^{1}$ for every $n\in\mathbb{N}$ as $k$ goes to $\infty$. By (1.15) it follows that $E(\mathcal{C}_{k}(\nu)(A_{n}))=\nu(A_{n})$. Thus $\mathcal{C}_{k}(\nu_{R})(A)=\sum_{n\in\mathbb{N}}\mathcal{C}_{k}(\nu|_{A_{n}})(X)$ converges in $\mathcal{L}^{1}$ to a limit $\mu_{\infty}(A)$ by Lemma 3.46. Thus $\mathcal{C}_{k}(\nu_{R})(A)$ converges in probability to $\mu_{\infty}(A)$ and $E(\mu_{\infty}(A))=\nu_{R}(A)$. Since $\mathcal{C}_{k}(\nu_{\perp})(A)\leq\mathcal{C}_{k}(\nu_{\perp})(X)$ we have that $\mathcal{C}_{k}(\nu_{\perp})(A)$ converges to [math] in probability. Thus $\mathcal{C}_{k}(\nu)(A)$ converges in probability to $\mu_{\infty}(A)$. ∎

Proposition 3.48**.**

Let $\nu$ be a finite Borel measure on $X$ such that $\nu(X\setminus X_{0})=0$. Assume that $\mathcal{C}_{k}(\nu|_{D})(X)=\mathcal{C}_{k}(\nu)(D)$ converges in $\mathcal{L}^{1}$ for every compact set $D\subseteq X_{0}$ with $I_{\varphi}(\nu|_{D})<\infty$. Assume that $\mathcal{C}_{k}(\nu_{\perp})(X)$ converges to [math] in probability. Then the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$.

Proof.

From the definition of $\mathcal{C}_{k}(\nu)$ it follows that that $E\left(\frac{\mathrm{d}\mathcal{C}_{k}(\nu)}{\mathrm{d}\nu}(x)\right)=1$ for every $x\in X_{0}$, i.e. for $\nu$ almost every $x\in X$. The assumptions of Theorem 3.15 are satisfied by Lemma 3.47 for $\mu_{k}=\mathcal{C}_{k}(\nu)$ and for the reference measure $\nu$. Hence Property i.), iii.), iv.) of Definition 1.11 hold for $\tau=\nu$ by Theorem 3.15 and Remark 3.16. Property v.), vi.) of Definition 1.11 hold for $\tau=\nu$ by Lemma 3.47, and Remark 3.16. Property vii.), viii.) of Definition 1.11 hold for $\tau=\nu$ by the fact that $\mathcal{C}_{k}(\nu_{\perp})(X)$ converges to [math] in probability. Property x.) of Definition 1.11 holds for $\tau=\nu$ by Proposition 3.20. Property xi.) of Definition 1.11 holds for $\tau=\nu$ by Proposition 3.18.

Now we show that Property ii.) of Definition 1.11 holds for $\tau=\nu$. Let $f:X\longrightarrow\mathbb{R}$ be a Borel measurable function such that $\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)<\infty$. It follows from Theorem 3.15 that $\intop_{X}f(x)\mathrm{d}\mathcal{C}_{k}(\nu)(x)$ converges to a random variable $S(f)$ in probability with $E(\left|S(f)\right|)\leq\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)$. Since v.) holds for $\tau=\nu$ it follows that $\nu_{R}(A)=E(\mathcal{C}(\nu)(A))$ for every Borel set $A\subseteq X$ and so $E(\intop_{X}f(x)\mathrm{d}\mathcal{C}(\nu)(x))=\intop_{X}f(x)\mathrm{d}\nu_{R}(x)$ holds when $f$ is a simple function and so by the monotone convergence theorem it holds for Borel measurable $f$. On the other hand $\intop_{X}f(x)\mathrm{d}\mathcal{C}(\nu)(x)=S(f)$ almost surely by iii.). Hence ii.) holds for $\tau=\nu$. In particular, applied to $\chi_{A}\cdot f$ for a nonnegative Borel function $f$ with $\intop_{X}f(x)\mathrm{d}\nu(x)<\infty$ and a Borel set $A\subseteq X$ we get that

[TABLE]

the sequence $\intop_{A}f(x)\mathrm{d}\mathcal{C}_{k}(\nu)(x)$ converges to $S(\chi_{A}f)$ in probability and $\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu)(x)=S(\chi_{A}f)$ almost surely. Moreover, $\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu)(x)=\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu_{R})(x)$ almost surely since Property viii.) of Definition 1.11 holds for $\tau=\nu$. There exists a Borel set $A_{0}\subseteq X$ such that $\nu_{R}(A_{0})=0$ and $\nu_{\perp}(X\setminus A_{0})=0$, then by Property *ii.) *of Definition 1.11 for $\tau=\nu$

[TABLE]

converges to $S(\chi_{A}\chi_{A_{0}}f)$ in probability and $E(S(\intop_{X}\chi_{A}\chi_{A_{0}}f))=\int_{A\cap A_{0}}f(x)\mathrm{d}\nu_{R}(x)=0$, so $S(\chi_{A}\chi_{A_{0}}f)=0$ almost surely since it is nonnegative. Hence $\intop_{A}f(x)\mathrm{d}\mathcal{C}_{k}(\nu_{R})(x)$ converges to $\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu)(x)=S(\chi_{A}f)=\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu_{R})(x)$ in $\mathcal{L}^{1}$ by Lemma 2.10 due to (3.5). In particular, if $\nu_{\perp}(A)=0$ for a Borel set $A\subseteq X$ then

[TABLE]

converges to $\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu)(x)=S(\chi_{A}f)=\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu_{R})(x)$ in $\mathcal{L}^{1}$.

Let $\tau\ll\nu$ be a deterministic finite Borel measure on $X$. If $I_{\varphi}(\tau|_{D})<\infty$ for a Borel set $D\subseteq X$ then $\tau|_{D}=\tau_{R}|_{D}$ and, by Proposition 1.9, there exists $A\subseteq D$ such that $\tau(D\setminus A)=0$ and $\nu_{\perp}(A)=0$. Thus, by the end of the last paragraph (3.7), we have that $\mathcal{C}_{k}(\tau)(D)=\intop_{D}\frac{\mathrm{d}\tau}{\mathrm{d}\nu}(x)\mathrm{d}\mathcal{C}_{k}(\nu)(x)=\intop_{A}\frac{\mathrm{d}\tau}{\mathrm{d}\nu}(x)\mathrm{d}\mathcal{C}_{k}(\nu)(x)$ converges to $\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu)(x)$ in $\mathcal{L}^{1}$. By (3.6) and Proposition 1.9 it follows that $\mathcal{C}_{k}(\tau_{\perp})(X)=\intop_{X}\frac{\mathrm{d}\tau}{\mathrm{d}\nu}(x)\mathrm{d}\mathcal{C}_{k}(\nu_{\perp})(x)$ converges to [math] in probability. Hence the assumptions of the statement not only hold for $\nu$ but the same assumptions also hold for $\tau$. Thus similarly to the way we proved that Property i.)-viii.), x.)-xi.) of Definition 1.11 hold for $\tau=\nu$, we can also prove the same properties for arbitrary $\tau\ll\nu$. Hence it is left to prove that ix.) holds.

Property ix.) of Definition 1.11 holds by Proposition 3.14 and the fact that the limit is unique by Proposition 3.5. ∎

Theorem 3.49**.**

Let $\nu$ be a finite Borel measure on $X$ such that $\nu(X\setminus X_{0})=0$. The conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ if and only if $\mathcal{C}_{k}(\nu|_{D})(X)=\mathcal{C}_{k}(\nu)(D)$ converges in $\mathcal{L}^{1}$ for every compact set $D\subseteq X_{0}$ with $I_{\varphi}(\nu|_{D})<\infty$ and $\mathcal{C}_{k}(\nu_{\perp})(X)$ converges to [math] in probability.

Proof.

Assume that the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$. Then by* *Property vii.) of Definition 1.11 it follows that $\mathcal{C}_{k}(\nu_{\perp})(X)$ converges to [math] in probability. Let $D\subseteq X_{0}$ be a compact set such that $I_{\varphi}(\nu|_{D})<\infty$. Then $\nu_{R}|_{D}=\nu|_{D}$ and so $\mathcal{C}_{k}(\nu)(D)=\mathcal{C}_{k}(\nu_{R})(D)$. Thus by Property vi.) of Definition 1.11 it follows that $\mathcal{C}_{k}(\nu)(D)$ converges to $\mathcal{C}(\nu)(D)$ in $\mathcal{L}^{1}$.

The other direction of the equivalence follows from Proposition 3.48. ∎

Lemma 3.50**.**

Let $\nu$ be a finite Borel measure on $X$ such that $\nu(X\setminus X_{0})=0$. Then $\mathcal{C}_{k}(\nu_{\perp})(X)$ converges to [math] in probability if and only if $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability for every compact set $D\subseteq X_{0}$.

Proof.

If $\mathcal{C}_{k}(\nu_{\perp})(X)$ converges to [math] in probability then of course $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability for every compact set $D\subseteq X$.

Assume that $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability for every compact set $D\subseteq X_{0}$. Let $\varepsilon>0$ be fixed. Let $D\subseteq X_{0}$ be a compact set such that $\nu(X\setminus D)<\varepsilon$, which we can choose by inner regularity. Then by the fact that $\rho(Y,Z)\leq E\left(\left|Y-Z\right|\right)$, by (1.15) and Lemma 3.13 it follows that

[TABLE]

[TABLE]

Hence

[TABLE]

because $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability. It holds for every $\varepsilon>0$ thus $\mathcal{C}_{k}(\nu_{\perp})(X)$ converges to [math] in probability. ∎

Remark 3.51*.*

Let $X$ be locally compact and let $\nu$ be a locally finite Borel measure on $X$ such that $\nu(X\setminus X_{0})=0$. Then $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability for every compact set $D\subseteq X_{0}$ if and only if $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability for every compact set $D\subseteq X$. It can easily be deduced from Lemma 3.50.

Proposition 3.52**.**

Let $X$ be locally compact and let $\nu$ be a locally finite Borel measure on $X$ such that $\nu(X\setminus X_{0})=0$. Assume that $\mathcal{C}_{k}(\nu|_{D})(X)=\mathcal{C}_{k}(\nu)(D)$ converges in $\mathcal{L}^{1}$ for every compact set $D\subseteq X_{0}$ with $I_{\varphi}(\nu|_{D})<\infty$. Assume that $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability for every compact set $D\subseteq X$. Then the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$.

Proof.

From the definition of $\mathcal{C}_{k}(\nu)$ it follows that that $E\left(\frac{\mathrm{d}\mathcal{C}_{k}(\nu)}{\mathrm{d}\nu}(x)\right)=1$ for every $x\in X_{0}$, i.e. for $\nu$ almost every $x\in X$. The assumptions of Theorem 3.41 are satisfied for $\mu_{k}=\mathcal{C}_{k}(\nu)$ and for the reference measure $\nu$ by Lemma 3.47 because $\nu|_{A}$ is a finite Borel measure for every Borel set $A\subseteq X$ with $\nu(A)<\infty$, in particular if $A$ is compact. Hence Property i.)*, iii.), iv.) of Definition 1.11 hold for $\tau=\nu$ by Theorem 3.41 and Remark 3.42. Property v.), vi.) of Definition 1.11 hold for $\tau=\nu$ by Lemma 3.47, and Remark 3.42. Property vii.), viii.) of Definition 1.11 hold for $\tau=\nu$ by the fact that

[TABLE]

converges to [math] in probability for every $f\in C_{c}(X)$. Property x.)* of Definition 1.11 holds for $\tau=\nu$ by Proposition 3.45. Property xi.) of Definition 1.11 holds for $\tau=\nu$ by Proposition 3.44.

Now we show that Property ii.) of Definition 1.11 hold for $\tau=\nu$. Let $f:X\longrightarrow\mathbb{R}$ be a Borel measurable function such that $\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)<\infty$. It follows from Theorem 3.41 that $\intop_{X}f(x)\mathrm{d}\mathcal{C}_{k}(\nu)(x)$ converges to a random variable $S(f)$ in probability with $E(\left|S(f)\right|)\leq\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)$. Since v.) holds for $\tau=\nu$ it follows that $\nu_{R}(A)=E(\mathcal{C}(\nu)(A))$ for every Borel set $A\subseteq X$ with $\nu(A)<\infty$ and so $E(\intop_{X}f(x)\mathrm{d}\mathcal{C}(\nu)(x))=\intop_{X}f(x)\mathrm{d}\nu_{R}(x)$ holds when $f$ is a simple function and so by the monotone convergence theorem it holds for Borel measurable $f$. On the other hand $\intop_{X}f(x)\mathrm{d}\mathcal{C}(\nu)(x)=S(f)$ almost surely by iii.). Hence ii.) holds for $\tau=\nu$. In particular, applied to $\chi_{A}\cdot f$ for a nonnegative, Borel measurable function $f$ and a Borel set $A\subseteq X$ with compact closure such that $\int_{A}f(x)\mathrm{d}\nu(x)<\infty$ we get that

[TABLE]

the sequence $\intop_{A}f(x)\mathrm{d}\mathcal{C}_{k}(\nu)(x)$ converges to $S(\chi_{A}f)$ in probability and $\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu)(x)=S(\chi_{A}f)$ almost surely. Moreover, $\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu)(x)=\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu_{R})(x)$ almost surely since Property viii.) of Definition 1.11 holds for $\tau=\nu$. There exists a Borel set $A_{0}\subseteq X$ such that $\nu_{R}(A_{0})=0$ and $\nu_{\perp}(X\setminus A_{0})=0$, then by Property *ii.) *of Definition 1.11 for $\tau=\nu$

[TABLE]

converges to $S(\chi_{A}\chi_{A_{0}}f)$ in probability and $E(S(\intop_{X}\chi_{A}\chi_{A_{0}}f))=\int_{A\cap A_{0}}f(x)\mathrm{d}\nu_{R}(x)=0$, so $S(\chi_{A}\chi_{A_{0}}f)=0$ almost surely since it is nonnegative. Hence $\intop_{A}f(x)\mathrm{d}\mathcal{C}_{k}(\nu_{R})(x)$ converges to $\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu)(x)=S(\chi_{A}f)=\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu_{R})(x)$ in $\mathcal{L}^{1}$ by Lemma 2.10 due to (3.8). In particular, if $\nu_{\perp}(A)=0$ for a Borel set $A\subseteq X$ with compact closure then

[TABLE]

converges to $\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu)(x)=S(\chi_{A}f)=\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu_{R})(x)$ in $\mathcal{L}^{1}$.

Let $\tau\ll\nu$ be a deterministic, locally finite, Borel measure on $X$. If $I_{\varphi}(\tau|_{D})<\infty$ for a compact set $D\subseteq X$ then $\tau|_{D}=\tau_{R}|_{D}$ and, by Proposition 1.9, there exists $A\subseteq D$ such that $\tau(D\setminus A)=0$ and $\nu_{\perp}(A)=0$. Thus, by the end of the last paragraph (3.10), we have that $\mathcal{C}_{k}(\tau)(D)=\intop_{D}\frac{\mathrm{d}\tau}{\mathrm{d}\nu}(x)\mathrm{d}\mathcal{C}_{k}(\nu)(x)=\intop_{A}\frac{\mathrm{d}\tau}{\mathrm{d}\nu}(x)\mathrm{d}\mathcal{C}_{k}(\nu)(x)$ converges to $\intop_{A}f(x)\mathrm{d}\mathcal{C}(\nu)(x)$ in $\mathcal{L}^{1}$. By (3.9) and Proposition 1.9 it follows that $\mathcal{C}_{k}(\tau_{\perp})(D)=\intop_{D}\frac{\mathrm{d}\tau}{\mathrm{d}\nu}(x)\mathrm{d}\mathcal{C}_{k}(\nu_{\perp})(x)$ converges to [math] in probability for every compact set $D\subseteq X$. Hence the assumptions of the statement not only hold for $\nu$ but the same assumptions also hold for $\tau$. Thus similarly to the way we proved that Property i.)-viii.), x.)-xi.) of Definition 1.11 hold for $\tau=\nu$, we can also prove the same properties for arbitrary $\tau\ll\nu$. Hence it is left to prove that ix.) holds.

Property ix.) of Definition 1.11 holds by Proposition 3.40 and the fact that the limit is unique by Proposition 3.33. ∎

Theorem 3.53**.**

Let $X$ be locally compact. Let $\nu$ be a locally finite Borel measure on $X$ such that $\nu(X\setminus X_{0})=0$. Then the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ if and only if $\mathcal{C}_{k}(\nu|_{D})(X)=\mathcal{C}_{k}(\nu)(D)$ converges in $\mathcal{L}^{1}$ for every compact set $D\subseteq X_{0}$ with $I_{\varphi}(\nu|_{D})<\infty$ and $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability for every compact set $D\subseteq X_{0}$.

Proof.

Assume that the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$. Then by* *Property ii.), iv.) and vii.) of Definition 1.11 it follows that $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability for every compact set $D\subseteq X_{0}$. Let $D\subseteq X_{0}$ be a compact subset such that $I_{\varphi}(\nu|_{D})<\infty$. Then $\nu_{R}|_{D}=\nu|_{D}$ and so $\mathcal{C}_{k}(\nu)(D)=\mathcal{C}_{k}(\nu_{R})(D)$. Thus by Property vi.) of Definition 1.11 it follows that $\mathcal{C}_{k}(\nu)(D)$ converges to $\mathcal{C}(\nu)(D)$ in $\mathcal{L}^{1}$.

The other direction of the equivalence follows from Proposition 3.52 and Remark 3.51. ∎

Proof of Proposition 1.14. By Theorem 3.49, Theorem 3.53 and Lemma 3.50 according to both of the two definition of the conditional measure in Definition 1.11 the conditional measure exists if and only if $\mathcal{C}_{k}(\nu|_{D})(X)=\mathcal{C}_{k}(\nu)(D)$ converges in $\mathcal{L}^{1}$ for every compact set $D\subseteq X_{0}$ with $I_{\varphi}(\nu|_{D})<\infty$ and $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability for every compact set $D\subseteq X$. The limit in Property* i.)* and in Property i.)* are the same almost surely because if $\mathcal{C}_{k}(\nu)$ weakly converges to $\mathcal{C}(\nu)$ subsequentially in probability then $\mathcal{C}_{k}(\nu)$ vaguely converges to $\mathcal{C}(\nu)$ in probability by Remark 3.31 and Proposition 3.23 and since the limit is unique by Proposition 3.33 the two limits are the same almost surely.$\hfill\square$

4 Decomposition of measure

We prove Proposition 1.7, Proposition 1.9 and Proposition 1.10 in this section.

Proposition 4.1**.**

Let $\nu$ be a finite, Borel measure on $X$. There exist two finite, Borel measures $\nu_{\varphi R}=\nu_{R}$ and $\nu_{\varphi\perp}=\nu_{\perp}$ with the following properties:

i) $\nu=\nu_{R}+\nu_{\perp}$

ii) $\nu_{R}\perp\nu_{\perp}$

iii) $\nu_{\perp}$ is singular to every finite Borel measure with finite $\varphi$-energy

iv) there exists a sequence of disjoint Borel sets $(A_{n})_{n\in\mathbb{N}}$ such that $\nu_{R}=\nu|_{\cup_{n\in\mathbb{N}}A_{n}}=\sum_{n\in\mathbb{N}}\nu|_{A_{n}}$ and $I_{\varphi}(\nu|_{A_{n}})=I_{\varphi}(\nu_{R}|_{A_{n}})<\infty$.

Proof.

Let

[TABLE]

We can find $A$ and $(A_{n})_{n\in\mathbb{N}}$ such that $A=\cup A_{n}$, $\nu(A)=c_{\max}$ and $I_{\varphi}(\nu|_{A_{n}})<\infty$ for all $n\in\mathbb{N}$. Without the loss of generality we can further assume that all the $A_{n}$ are disjoint for different $n$. Let $\nu_{R}=\nu|_{A}$ and $\nu_{\bot}=\nu|_{X\setminus A}$. Then i), ii) and iv) are satisfied. Assume for a contradiction that iii) is not satisfied that is there exists a Borel probability measure $\tau$ with $I_{\varphi}(\tau)<\infty$ such that $\tau\ll\nu_{\bot}$. Then there exists $N>0$ such that $\nu_{\bot}(C_{N})>0$ where $C_{N}=\left\{x:\frac{\mathrm{d}\tau}{\mathrm{d}\nu_{\bot}}(x)\geq\frac{1}{N}\right\}$. If $D\subseteq C_{N}$ then

[TABLE]

Thus $I_{\varphi}(\nu_{\bot}|_{C_{N}})\leq N^{2}I_{\varphi}(\tau|_{C_{N}})\leq N^{2}I_{\varphi}(\tau)<\infty$. This contradicts with the maximality of $c_{\max}$. ∎

Proposition 4.2**.**

Let $\nu$ be a finite Borel measure on $X$ and let $\tau\ll\nu$ be a finite Borel measure on $X$. Then $\frac{\mathrm{d}\tau}{\mathrm{d}\nu}(x)\mathrm{d}\nu_{\varphi R}=\tau_{\varphi R}$ and $\frac{\mathrm{d}\tau}{\mathrm{d}\nu}(x)\mathrm{d}\nu_{\varphi\perp}=\tau_{\varphi\perp}$.

Proof.

It is easy to see that the statement holds for $\tau|_{A_{n}}$ in place of $\tau$ where $A_{n}=\left\{x\in X:1/n<\frac{\mathrm{d}\tau}{\mathrm{d}\nu}(x)<n\right\}$ and so holds for $\tau|_{\cup_{n\in\mathbb{N}}A_{n}}$. Since $\tau$ is a finite Borel measure $\tau(X\setminus\cup_{n\in\mathbb{N}}A_{n})=0$ and so the statement holds. ∎

Remark 4.3*.*

Proposition 4.1 and Proposition 4.2 hold for locally finite, Borel measures. This can easily be deduced from Proposition 4.1, Proposition 4.2 and the Lindelöf property of $X$.

Proposition 4.4**.**

Let $\tau$ be a finite Borel measure on $X$ such that $\intop\intop\varphi(x,y)\mathrm{d}\tau(y)\mathrm{d}\tau(x)<\infty$. Then for every $\varepsilon>0$ there exist $0<M<\infty$ and a compact set $F\subseteq X$ such that $\tau(X\setminus F)<\varepsilon$ and $\intop_{F}\varphi(x,y)\mathrm{d}\tau(y)<M$ for every $x\in X$.

Proof.

Since $\intop\intop\varphi(x,y)\mathrm{d}\tau(y)\mathrm{d}\tau(x)<\infty$ it follows that $\intop\varphi(x,y)\mathrm{d}\tau(y)<\infty$ for $\tau$ almost every $x\in X$. Thus there exists a compact set $F_{0}\subseteq X$ and $M_{0}>0$ such that $\tau(X\setminus F_{0})<\varepsilon/2$ and $\intop\varphi(x,y)\mathrm{d}\tau(y)<M_{0}$ for every $x\in F_{0}$. In the proof of [3, Chapter III. Thm 1, page 15] it is shown that there exists a compact set $F\subseteq F_{0}$ such that $\tau(X\setminus F)<\varepsilon$ and $\lim_{x\rightarrow x_{0}}\intop_{F}\varphi(x,y)\mathrm{d}\tau(y)=\intop_{F}\varphi(x_{0},y)\mathrm{d}\tau(y)<M_{0}$ for every $x_{0}\in F$. Hence there exists $r>0$ such that if $\mathrm{dist}(F,x)<r$ then $\intop_{F}\varphi(x,y)\mathrm{d}\tau(y)<M_{0}$. Whenever $\mathrm{dist}(F,x)\geq r$ then $\intop_{F}\varphi(x,y)\mathrm{d}\tau(y)\leq\varphi(r)\tau(X)$ because $\varphi$ is monotone decreasing. Hence the statement follows. ∎

Proposition 4.5**.**

If $\nu$ is a finite Borel measure that is singular to every finite Borel measure with finite $\varphi$-energy then there exits a Borel set $Z\subseteq X$ such that $\nu(X\setminus Z)=0$ and $C_{\varphi}(Z)=0$.

Proof.

Let $A_{n}=\left\{x\in X:\int\varphi(x,y)\mathrm{d}\nu(y)\leq n\right\}$. Then

[TABLE]

Thus $\nu(A_{n})=0$ by the assumption and hence $\nu(X\setminus Z)=0$ for $Z=\left\{x\in X:\int\varphi(x,y)\mathrm{d}\nu(y)=\infty\right\}$.

Assume for a contradiction that $C_{\varphi}(Z)>0$. Then there exists a probability measure $\tau$ on $Z$ such that $\intop\intop\varphi(x,y)\mathrm{d}\tau(y)\mathrm{d}\tau(x)<\infty$. Then by Proposition 4.4 there exist $0<M<\infty$ and $F\subseteq Z$ such that $\tau(F)>0$ and $\intop_{F}\varphi(x,y)\mathrm{d}\tau(y)<M$ for every $x\in X$. Thus

[TABLE]

contradicting with that

[TABLE]

where we used Fubini‘s theorem. Hence $C_{\varphi}(Z)=0$. ∎

Remark 4.6*.*

Proposition 4.5 holds for locally finite, Borel measures. This can easily be deduced from Proposition 4.5, the Lindelöf property of $X$ and the fact that $C_{\varphi}(\cup_{n=1}^{\infty}A_{n})=0$ for a sequence of Borel sets $A_{n}$ with $C_{\varphi}(A_{n})=0$.

5 Degenerate case

In this section we discuss why the conditional measure of the $\varphi$-singular part $\nu_{\perp}$ vanishes.

Lemma 5.1**.**

Let $D\subseteq X$ be a compact set and let $B_{\omega}$ be a closed realisation of the random closed set $B$ such that $D\cap B_{\omega}=\emptyset$. Let $\nu$ be a finite Borel measure such that $\mathrm{supp}\nu\subseteq D$. Then $\mathcal{C}_{k}(\nu)(X)=\mathcal{C}_{k,\omega}(\nu)(X)$ converges to [math] for that realisation $B_{\omega}$.

Proof.

Since $B\cap D=\emptyset$ then $\mathrm{dist}(B,D)>0$. Let $k_{0}$ be such that $\sup\left\{\mathrm{diam}(Q):Q\in\mathcal{Q}_{k}\right\}<\mathrm{dist}(B,D)$ for ever $k\geq k_{0}$ (we note that $k_{0}$ depends on the realisation $B_{\omega}$ but exists nevertheless). Then $\mathcal{C}_{k,\omega}(\nu)(X)=0$ by the definition of $\mathcal{C}_{k}$ for $k\geq k_{0}$. ∎

Theorem 5.2**.**

Assume that if $C_{\varphi}(D)=0$ for some compact set $D\subseteq X$ then $B\cap D=\emptyset$ almost surely. If $\nu$ is a finite Borel measure that is singular to every finite Borel measure with finite $\varphi$-energy then $\mathcal{C}_{k}(\nu)(X)$ converges to [math] in probability.

Proof.

By Proposition 4.5 there exists $Z\subseteq X$ such that $\nu(X\setminus Z)=0$ and $C_{\varphi}(Z)=0$. Let $\eta>0$ be fixed. There exists a compact set $D\subseteq Z$ such that $\nu(X\setminus D)<\eta$ by inner regularity. Then by assumption $B\cap D=\emptyset$ almost surely. Thus $\mathcal{C}_{k}(\nu|_{D})(X)$ converges to [math] almost surely by Lemma 5.1 and hence converges to [math] in probability. Since $\mathcal{C}_{k}(\nu)=\mathcal{C}_{k}(\nu|_{D})+\mathcal{C}_{k}(\nu|_{X\setminus D})$ it follows by (1.15) that

[TABLE]

[TABLE]

Since we can choose $\eta$ to be arbitrarily small it follows that $\mathcal{C}_{k}(\nu)(X)$ converges to [math] in probability. ∎

Remark 5.3*.*

Let $\nu$ be such that $\nu(X\setminus X_{0})=0$. Then for the conclusion of Theorem 5.2 it is enough to assume that if $C_{\varphi}(D)=0$ for some compact set $D\subseteq X_{0}$ (rather than for compact sets $D\subseteq X$) then $B\cap D=\emptyset$ almost surely. In the proof we can choose $D\subseteq Z\cap X_{0}$.

Lemma 5.4**.**

Let $\nu$ be a finite Borel measure on $X$ such that $\nu(X_{0})=0$. Then $\mathcal{C}_{k}(\nu)(X)$ converges to [math] in $\mathcal{L}^{1}$ and so in probability.

Proof.

It follows from definition $\mathcal{C}_{k}(\nu)$ that

[TABLE]

Since $\nu(X_{0})=0$ it follows that $\lim_{k\rightarrow\infty}\nu(\bigcup_{\mathcal{Q}_{k}}Q)=0$ and so the statement follows. ∎

Remark 5.5*.*

Let $X$ be locally compact and $\nu$ be a locally finite Borel measure. If $\nu(X_{0})=0$ then $\mathcal{C}_{k}(\nu)(K)$ converges to [math] in $\mathcal{L}^{1}$ and so in probability for every compact set $K$. It follows easily from Lemma 5.4. That implies that $\mathcal{C}_{k}(\nu)(K)$ converges to [math] vaguely in probability.

6 $\mathcal{L}^{2}$-boundedness

In this section we show that under the assumptions of Section 1.1.2 if $\nu$ is a finite Borel measure with $I_{\varphi}(\nu)<\infty$ then the sequence $\mathcal{C}_{k}(\nu)(X)$ is $\mathcal{L}^{2}$-bounded.

Lemma 6.1**.**

Let $Q,S\in\mathcal{Q}_{k}$, $Q\neq S$ and $\nu_{1}$ and $\nu_{2}$ be finite Borel measures on $X$. If (1.23) holds for $Q$ and $S$ then

[TABLE]

Proof.

By the definition of $\mathcal{C}_{k}(\nu)$ and (1.23) it follows that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

∎

Lemma 6.2**.**

Let $\nu_{1}$ and $\nu_{2}$ be finite Borel measures on $X$. Assume that $\delta>0$ is such that (1.16) holds and $Q,S\in\mathcal{Q}_{k}$ such that $\max\left\{\mathrm{diam}(Q),\mathrm{diam}(S)\right\}<\delta\cdot\mathrm{dist}(Q,S)$. Then

[TABLE]

Proof.

If $x\in Q$ and $y\in S$ then

[TABLE]

Then by (1.16) and the monotonicity of $\varphi$ it follows that

[TABLE]

Integrating over $Q\times S$ with respect to $\nu_{1}\times\nu_{2}$ the statement follows. ∎

Lemma 6.3**.**

Assume that (1.22) holds. Let $\nu_{1}$ and $\nu_{2}$ be finite Borel measures on $X$. If $Q,S\in\mathcal{Q}_{k}$ then

[TABLE]

Proof.

Due to symmetry without the loss of generality we can assume that $\nu_{1}(Q)\leq\nu_{2}(S)$. Also we can assume that $0<\nu_{1}(Q)\leq\nu_{2}(S)$ otherwise the proof is trivial. By the definition of $\mathcal{C}_{k}$ and (1.22) it follows that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

∎

Proposition 6.4**.**

Let $\nu$ be a finite Borel measure on $X$. Assume that (1.22) holds and there exists $\delta>0$ such that (1.16), (1.23) and (1.18) hold. Then

[TABLE]

for every $k\in\mathbb{N}$.

Proof.

We say that a pair $(Q,S)\in\mathcal{Q}_{k}\times\mathcal{Q}_{k}$ is a ‘good’ pair if $\max\left\{\mathrm{diam}(Q),\mathrm{diam}(S)\right\}<\delta\cdot\mathrm{dist}(Q,S)$ and is a bad pair if $\max\left\{\mathrm{diam}(Q),\mathrm{diam}(S)\right\}\geq\delta\cdot\mathrm{dist}(Q,S)$. Combining Lemma 6.1 and Lemma 6.2 it follows that

[TABLE]

[TABLE]

[TABLE]

By Lemma 6.3 and (1.18)

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Hence the statement follows by

[TABLE]

[TABLE]

[TABLE]

∎

7 Non-degenerate limit

In this section our main goal is to show that $\mathcal{C}_{k}(\nu)(A)$ converges in $\mathcal{L}^{2}$ if $I_{\varphi}(\nu)<\infty$ and $\underline{F}(x,y)=\overline{F}(x,y)$. Recall that the definition of $F_{k,n}$, $\underline{F}(x,y)$ and $\overline{F}(x,y)$ can be found in Section 1.3. A key observation is that $E\left(\mathcal{C}_{k}(\nu)(X)^{2}\right)=\intop_{X}\intop_{X}F_{k,k}(x,y)\mathrm{d}\nu(x)\mathrm{d}\nu(y)$. Using this and the assumption that $\underline{F}(x,y)=\overline{F}(x,y)$ we prove the $\mathcal{L}^{2}$ convergence in two steps. We divide the double integral into two parts, one part is the double integral on a domain that is bounded away from the diagonal and approximates the double integral uniformly, the other part is around the diagonal that is small. Then from this we deduce the convergence in $\mathcal{L}^{2}$. At the end of the section we show that if $\mathcal{C}_{k}(\nu)(A)$ is a martingale then we do not even need the assumption that $\underline{F}(x,y)=\overline{F}(x,y)$ because then the convergence in $\mathcal{L}^{2}$ is automatic by the $\mathcal{L}^{2}$-boundedness.

Lemma 7.1**.**

Let $\nu_{1}$ and $\nu_{2}$ be finite Borel measures on $X$. For $k,n\in\mathbb{N}$

[TABLE]

Proof.

By the definition of $\mathcal{C}_{k}$ it follows that

[TABLE]

[TABLE]

∎

Lemma 7.2**.**

Let $\nu_{1}$ and $\nu_{2}$ be finite Borel measures on $X$. Let $Q,S\subseteq X$ be Borel sets. For $k,n\in\mathbb{N}$, it follows that

[TABLE]

Lemma 7.2 follows by the application of Lemma 7.1 to the measures $\nu_{1}|_{Q}$ and $\nu_{2}|_{S}$.

Lemma 7.3**.**

Let $\nu_{1}$ and $\nu_{2}$ be finite Borel measures on $X$. Assume that there exists $\delta>0$ such that (1.23) holds. Let $\varepsilon,\eta>0$ and let $A_{\varepsilon}=\left\{(x,y)\in X\times X:d(x,y)>\varepsilon\right\}$. Then there exists $m\in\mathbb{N}$ such that for every $n,k\geq m$

[TABLE]

Proof.

Let $k_{0}\in\mathbb{N}$ be large enough that $\sup\left\{\mathrm{diam}(Q):Q\in\mathcal{Q}_{k}\right\}<\varepsilon/3$ for every $k>k_{0}$, we can choose such $k_{0}$ due to (1.10). Whenever $A_{\varepsilon}\cap Q\times S\neq\emptyset$ for $Q\in\mathcal{Q}_{k}$, $S\in\mathcal{Q}_{n}$, $k,n\geq k_{0}$ then $\mathrm{dist}(Q,S)>\varepsilon/3$. We can choose $k_{1}\geq k_{0}$ such that $\sup\left\{\mathrm{diam}(Q):Q\in\mathcal{Q}_{k}\right\}<\delta\varepsilon/3$ for every $k\geq k_{1}$. Whenever $A_{\varepsilon}\cap Q\times S\neq\emptyset$ for $Q\in\mathcal{Q}_{k}$, $S\in\mathcal{Q}_{n}$, $k,n\geq k_{1}$ then $\max\left(\mathrm{diam}(Q),\mathrm{diam}(S)\right)<\delta\varepsilon/3<\delta\mathrm{dist}(Q,S)$. Hence $F_{k,n}(x,y)\leq c\varphi(\mathrm{dist}(Q,S))$ for every $(x,y)\in Q\times S$ by (1.23). Since $\varphi$ is a nonnegative, monotone decreasing, continuous function it follows that $\varphi$ is absolutely continuous on $[\varepsilon/3,\infty)$, hence we can choose $k_{2}\geq k_{1}$, due to (1.10), such that $\varphi(\mathrm{dist}(Q,S))\leq\varphi(d(x,y))+\eta/c$ whenever $A_{\varepsilon}\cap Q\times S\neq\emptyset$ for $x\in Q\in\mathcal{Q}_{k}$, $y\in S\in\mathcal{Q}_{n}$, $k,n\geq k_{2}$. Hence

[TABLE]

whenever $A_{\varepsilon}\cap Q\times S\neq\emptyset$ for $x\in Q\in\mathcal{Q}_{k}$, $y\in S\in\mathcal{Q}_{n}$ $k,n\geq k_{2}$. Thus

[TABLE]

for $(x,y)\in A_{\varepsilon}$, $N\geq k_{2}$ and $k,n\geq N$. Since $\underline{F_{N}}$ and $\overline{F_{N}}$ converge as $N$ goes to $\infty$, due to the dominated convergence theorem, there exists $m\geq k_{2}$ such that

[TABLE]

and

[TABLE]

for every $N\geq m$. Thus the statement follows. ∎

Notation 7.4*.*

Assume that (1.19) holds. For $\varepsilon>0$ let $k_{\varepsilon}$ be the largest positive integer such that $\inf\left\{\mathrm{diam}(Q):Q\in\mathcal{Q}_{k_{\varepsilon}}\right\}>\varepsilon$, it is well-defined by (1.19) and (1.10) if $\varepsilon>0$ is small enough. Let $r_{\varepsilon}=\sup\left\{\mathrm{diam}(Q):Q\in\mathcal{Q}_{k_{\varepsilon}}\right\}$. Note that $r_{\varepsilon}$ converges to [math] as $\varepsilon$ approaches [math] by (1.10) and (1.19).

Lemma 7.5**.**

Let $\nu_{1}$ and $\nu_{2}$ be finite Borel measures on $X$ and let $\nu=\nu_{1}+\nu_{2}$. Let $\varepsilon>0$, let $G_{\varepsilon}=\left\{(x,y)\in X\times X:d(x,y)\leq\varepsilon\right\}$ and $H_{\varepsilon}=\left\{(x,y)\in X\times X:d(x,y)\leq\varepsilon+2r_{\varepsilon}\right\}$. Assume that (1.19) and (1.22) hold and there exists $0<\delta<1$ such that (1.16), (1.23) and (1.18). Then there exists $c_{4}>0$, depending on $a$, $c$, $c_{2}$, $c_{3}$ and $M_{\delta}$, such that

[TABLE]

for every $n\in\mathbb{N}$.

Proof.

Whenever $Q\times S\cap G_{\varepsilon}\neq\emptyset$ for $Q,S\in\mathcal{Q}_{k_{\varepsilon}}$ then $\mathrm{dist}(Q,S)\leq\varepsilon<\max\left\{\mathrm{diam}(Q),\mathrm{diam}(F)\right\}$. In particular, $\delta\cdot\mathrm{dist}(Q,S)<\max\left\{\mathrm{diam}(Q),\mathrm{diam}(F)\right\}$. Hence

[TABLE]

for every $Q\in\mathcal{Q}_{k_{\varepsilon}}$ by (1.18). By Lemma 7.2 and Lemma 6.4 it follows that

[TABLE]

[TABLE]

[TABLE]

Whenever $Q\times S\cap G_{\varepsilon}\neq\emptyset$ for $Q,S\in\mathcal{Q}_{k_{\varepsilon}}$ then $Q\times S\subseteq H_{\varepsilon}$. Hence

[TABLE]

By (7.1)

[TABLE]

[TABLE]

and similarly

[TABLE]

It is easy to see that

[TABLE]

Since

[TABLE]

and

[TABLE]

the statement follows by combining (7.2), (7.3), (7.4), (7.5) and (7.6). ∎

7.1 Limit in the presence of the kernel $F(x,y)$

Lemma 7.6**.**

Let $\nu$ be finite Borel measure. Then

[TABLE]

for every $k,n\in\mathbb{N}$. If

[TABLE]

then $\mathcal{C}_{k}(\nu)(X)$ converges in $\mathcal{L}^{2}$.

Proof.

Since $\mathcal{L}^{2}$ is complete it is sufficient to show that $\mathcal{C}_{k}(\nu)(X)$ is a Cauchy sequence in $\mathcal{L}^{2}$. For the integers $n\leq k$ by Lemma 7.1

[TABLE]

[TABLE]

hence (7.7) holds and by the assumption

[TABLE]

and so $\mathcal{C}_{k}(\nu)(X)$ is a Cauchy sequence in $\mathcal{L}^{2}$. ∎

Proposition 7.7**.**

Let $\nu$ be a finite Borel measure on $X$ with $I_{\varphi}(\nu)<\infty$. Assume that (1.17), (1.19) and (1.22) hold and there exists $0<\delta<1$ such that (1.16), (1.23) and (1.18) hold. If $\underline{F}(x,y)=\overline{F}(x,y)$ for $\nu\times\nu$ almost every $(x,y)$ then

[TABLE]

Proof.

Let $\eta,\varepsilon>0$. Let $m\in\mathbb{N}$ be as in Lemma 7.3 and $k,n\geq m$. Let $A_{\varepsilon}$, $G_{\varepsilon}$ and $H_{\varepsilon}$ as in Lemma 7.3 and Lemma 7.5. Then

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where we used Lemma 7.5 for $\nu_{1}=\nu_{2}=\nu$ and the fact that $\underline{F}(x,y)=\overline{F}(x,y)$ for $\nu\times\nu$ almost every $(x,y)$. Thus

[TABLE]

[TABLE]

By (1.17) and Fubini‘s theorem $\nu\times\nu\left((x,x):x\in X\right)=0.$ Since $\nu\times\nu$ and $\varphi(x,y)\mathrm{d}\nu(x)\mathrm{d}\nu(y)$ are finite measures it follows that

[TABLE]

converges to [math] as $\varepsilon$ approaches [math] because $r_{\varepsilon}$ goes to [math] by Notation 7.4. Taking the limit as $\eta$ and $\varepsilon$ go to [math] in (7.8) the statement follows by (7.7). ∎

Theorem 7.8**.**

Let $\nu$ be a finite Borel measure on $X$ such that $\nu(X\setminus X_{0})=0$ and $I_{\varphi}(\nu)<\infty$. Assume that (1.17), (1.19) and (1.22) hold and there exists $0<\delta<1$ such that (1.16), (1.23) and (1.18) hold. If $\underline{F}(x,y)=\overline{F}(x,y)$ for $\nu\times\nu$ almost every $(x,y)$ then for every Borel set $A\subseteq X$ it follows that $\mathcal{C}_{k}(\nu)(A)$ converges to a limit $\mu(A)$ in $\mathcal{L}^{2}$, in $\mathcal{L}^{1}$ and in probability and $E(\mu(A))=E(\mathcal{C}_{k}(\nu)(A))=\nu(A)$.

Proof.

By applying Lemma 7.6 and Proposition 7.7 to the measure $\nu|_{A}$ it follows that $\mathcal{C}_{k}(\nu)(A)$ converges to a limit $\mu(A)$ in $\mathcal{L}^{2}$ and so in $\mathcal{L}^{1}$ and in probability. Thus by (1.15) we have that $E(\mu(A))=E(\mathcal{C}_{k}(\nu)(A))=\nu(A)$. ∎

7.2 Limit in the presence of a martingale filtration

Theorem 7.9**.**

Let $\nu$ be a finite Borel measure on $X$ such that $\nu(X\setminus X_{0})=0$. Assume that for a Borel set $A\subseteq X$ there exists a filtration $\mathcal{F}_{k}$ such that $\mathcal{C}_{k}(\nu)(A)$ is a martingale with respect to the filtration $\mathcal{F}_{k}$. Then $\mathcal{C}_{k}(\nu)(A)$ converges to a limit $\mu(A)$ almost surely and $E(\mu(A))\leq E(\mathcal{C}_{k}(\nu)(A))=\nu(A)$.

If additionally $I_{\varphi}(\nu)<\infty$, (1.22) holds and there exists $\delta>0$ such that (1.16), (1.23) and (1.18) hold then $\mathcal{C}_{k}(\nu)(A)$ converges in $\mathcal{L}^{2}$, in $\mathcal{L}^{1}$ and $E(\mu(A))=E(\mathcal{C}_{k}(\nu)(A))=\nu(A)$.

Proof.

Since $\mathcal{C}_{k}(\nu)(A)$ is a nonnegative martingale it converges almost surely to a random limit $\mu(A)$ and $E(\mu(A))\leq E(\mathcal{C}_{k}(\nu)(A))=\nu(A)$ by the nonnegative martingale limit theorem [8, Theorem 5.2.9]. If $I_{\varphi}(\nu)<\infty$ then $\mathcal{C}_{k}(\nu)(A)$ is $\mathcal{L}^{2}$-bounded by Proposition 6.4, hence converges in $\mathcal{L}^{2}$ and in $\mathcal{L}^{1}$ by [8, Theorem 5.4.5]. Thus by (1.15) we have that $E(\mu(A))=E(\mathcal{C}_{k}(\nu)(A))=\nu(A)$. ∎

8 Existence of the conditional measure

We are now prepared to prove the existence of the conditional measure $\mathcal{C}(\nu)$ when $F(x,y)=\underline{F}(x,y)=\overline{F}(x,y)$.

Theorem 8.1**.**

Let $\nu$ be a finite, Borel measure on $X$ or let $X$ be locally compact and $\nu$ be a locally finite, Borel measure on $X$. Assume that $\nu(X\setminus X_{0})=0$. Assume that if $C_{\varphi}(D)=0$ for some compact set $D\subseteq X_{0}$ then $B\cap D=\emptyset$ almost surely. Assume that (1.17), (1.19) and (1.22) hold and there exists $0<\delta<1$ such that (1.16), (1.23) and (1.18) hold. Assume that at least one of the following conditions hold:

A.) $\underline{F}(x,y)=\overline{F}(x,y)$ for $\nu\times\nu$ almost every $(x,y)$,

B.) for every Borel set $A\subseteq X$ with $\nu(A)<\infty$ there exists a filtration $\mathcal{F}_{k}$ such that $\mathcal{C}_{k}(\nu)(A)$ is a martingale with respect to the filtration $\mathcal{F}_{k}$.

Then the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$. Moreover, if $A\subseteq X$ is a Borel set such that $\nu(A)<\infty$ and $I_{\varphi}(\nu|_{A})<\infty$ then $\mathcal{C}_{k}(\nu)(A)$ converges to $\mathcal{C}(\nu)(A)$ in $\mathcal{L}^{2}$.

Proof.

By Theorem 7.8 and Theorem 7.9 if $D\subseteq X$ is a Borel set such that $I_{\varphi}(\nu|_{D})<\infty$ and $\nu(D)<\infty$ then $\mathcal{C}_{k}(\nu)(D)$ converges in $\mathcal{L}^{2}$ and in $\mathcal{L}^{1}$ to a random variable $\mu(D)$. By Theorem 5.2 and Remark 5.3 we have that $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability for every compact set $D\subseteq X$. Thus it follows by Theorem 1.16 that the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$.

Let $A\subseteq X$ be a Borel set such that $\nu(A)<\infty$ and $I_{\varphi}(\nu|_{A})<\infty$. As we established at the beginning of the proof $\mathcal{C}_{k}(\nu)(A)$ converges to $\mu(A)$ in $\mathcal{L}^{2}$. By Property iv.) of Definition 1.11 it follows that $\mu(A)=\mathcal{C}(\nu)(A)$ almost surely. Hence it follows that $\mathcal{C}_{k}(\nu)(A)$ converges to $\mathcal{C}(\nu)(A)$ in $\mathcal{L}^{2}$. ∎

Remark 8.2*.*

If we assume that $\nu_{R}=\nu$ in Theorem 8.1 then for the conclusion to hold we do not even need the assumption that if $C_{\varphi}(D)=0$ for some compact set $D\subseteq X_{0}$ then $B\cap D=\emptyset$ almost surely. It is because we only use this assumption to ensure that $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability. However, if $\nu_{R}=\nu$ then $\mathcal{C}_{k}(\nu_{\perp})(X)=0$ for every $k$.

9 Double integration

In this section we prove the double integration formula (1.24).

Proposition 9.1**.**

Assume that (1.17), (1.19) and (1.22) hold and there exists $0<\delta<1$ such that (1.16), (1.23) and (1.18) hold. Let $\nu_{1}$ and $\nu_{2}$ be finite Borel measures on $X$ with $I_{\varphi}(\nu_{1}+\nu_{2})<\infty$. Assume that the conditional measure $\mathcal{C}(\nu_{i})$ of $\nu_{i}$ on $B$ exist with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ for $i=1,2$. Assume that if $A\subseteq X$ is a Borel set then $\mathcal{C}_{k}(\nu_{i})(A)$ converges to $\mathcal{C}(\nu_{i})(A)$ in $\mathcal{L}^{2}$ for $i=1,2$. Then

[TABLE]

[TABLE]

for every Borel sets $A_{1},A_{2}\subseteq X$.

Proof.

Without the loss of generality we can assume that $A_{1}$ and $A_{2}$ are both $X$ by restricting the measures to $\nu_{1}|_{A_{1}}$ and $\nu_{2}|_{A_{2}}$. We have that $\mathcal{C}_{k}(\nu_{1})(X)$ converges to $\mathcal{C}(\nu_{1})(X)$ in in $\mathcal{L}^{2}$ and $\mathcal{C}_{k}(\nu_{2})(X)$ converges to $\mathcal{C}(\nu_{2})(X)$ in $\mathcal{L}^{2}$. Thus $\mathcal{C}_{k}(\nu_{1})(X)\cdot\mathcal{C}_{k}(\nu_{2})(X)$ converges to $\mathcal{C}(\nu_{1})(X)\cdot\mathcal{C}(\nu_{2})(X)$ in $\mathcal{L}^{1}$ and in particular,

[TABLE]

by Lemma 7.1.

Let $\varepsilon,\eta>0$ be fixed. Then, by Lemma 7.3 and by Lemma 7.5, for large enough $k\in\mathbb{N}$

[TABLE]

[TABLE]

Let $D=\left\{(x,x):x\in X\right\}$. Then $\nu(D)=0$ by (1.17) and Fubini‘s theorem. Hence by the fact that $\intop_{X}\intop_{X}\varphi(x,y)\mathrm{d}\nu(x)\mathrm{d}\nu(y)<\infty$ it follows that

[TABLE]

by Notation 7.4. Similarly

[TABLE]

Hence by (9.2), (9.3), (9.4), (9.5) and Remark 1.20

[TABLE]

By Lemma 7.3, for large enough $k\in\mathbb{N}$

[TABLE]

By Remark 1.20 we have that $\underline{F}(x,y)\leq c\cdot\varphi(x,y)$ for $\nu\times\nu$ almost every $(x,y)$. Thus similarly to (9.4)

[TABLE]

Then it follows from (9.2), (9.7) and (9.8) that

[TABLE]

So the statement follows from (9.6) and (9.9). ∎

Proposition 9.2**.**

Assume that (1.17), (1.19) and (1.22) hold and there exists $0<\delta<1$ such that (1.16), (1.23) and (1.18) hold. Let $\nu_{1}$ and $\nu_{2}$ be finite Borel measures on $X$ with $I_{\varphi}(\nu_{1}+\nu_{2})<\infty$. Assume that the conditional measure $\mathcal{C}(\nu_{i})$ of $\nu_{i}$ on $B$ exist with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ for $i=1,2$. Assume that if $A\subseteq X$ is a Borel set then $\mathcal{C}_{k}(\nu_{i})(A)$ converges to $\mathcal{C}(\nu_{i})(A)$ in $\mathcal{L}^{2}$ for $i=1,2$. Let $f:X\times X\longrightarrow\mathbb{R}$ be a nonnegative Borel function. Then

[TABLE]

[TABLE]

Proof.

It follows from Proposition 9.1 that (9.10) holds for functions of the form $f(x,y)=\chi_{A_{1}}(x)\cdot\chi_{A_{2}}(y)$ for Borel sets $A_{1},A_{2}\subseteq X$. Hence, by the fact that the sets of the form $A_{1}\times A_{2}$ form a semi-ring generating the Borel $\sigma$-algebra of $X\times X$ we can deduce that (9.10) holds for $f(x,y)=I_{A}(x,y)$ for Borel sets $A\subseteq X\times X$ by Proposition 2.33. It follows that (9.10) holds for non-negative simple functions on $X\times X$ and so we can deduce (9.10) for every nonnegative Borel function on $X\times X$ using the monotone convergence theorem. ∎

Lemma 9.3**.**

Let $\nu$ and $\tau$ be locally finite Borel measures on $X$. Then there exist a sequence of finite measures $\left\{\nu_{i}\right\}_{i=1}^{\infty}$ and another sequence of finite measures $\left\{\tau_{i}\right\}_{i=1}^{\infty}$ such that $\nu_{R}=\sum_{i=1}^{\infty}\nu_{i}$, $\tau_{R}=\sum_{i=1}^{\infty}\tau_{i}$ and $I_{\varphi}(\nu_{i}+\tau_{i})<\infty$ for every $i,j\in\mathbb{N}$.

Proof.

By the Lebesgue decomposition [8, Theorem A.4.5] there exist a nonnegative Borel function $g$ and a locally finite Borel measure $\nu_{s}$ such that

[TABLE]

and $\nu_{s}$ is singular to $\tau_{R}$. Let $G\subseteq X$ be a Borel set such that $\tau_{R}(X\setminus G)=0$ and $\nu_{s}(G)=0$. Since $\nu_{R}$ is locally finite and $X$ is separable metric space it follows that $g(x)<\infty$ for $\tau_{R}$ almost all $x$. Let $G_{i}=\left\{x\in G:i-1\leq g(x)<i\right\}$ for every $i\in\mathbb{N}$. By Proposition 1.7 for every $i\in\mathbb{N}$ we can find a sequence $\left\{E_{i,j}\right\}_{j=1}^{\infty}$ of disjoint Borel subsets of $G_{i}$ such that $I_{\varphi}(\tau_{R}|_{E_{i,j}})<\infty$ for every $j\in\mathbb{N}$ and $\tau_{R}|_{G_{i}}=\sum_{j=1}^{\infty}\tau_{R}|_{E_{i,j}}$. By Lemma 2.12 and Remark 2.13 we can further assume that $\left\{E_{i,j}\right\}_{j=1}^{\infty}$ is a collection of disjoint compact sets and so $\tau_{R}(E_{i,j})<\infty$ for every $j\in\mathbb{N}$. Similarly, we can also find a collection of disjoint compact sets $\left\{A_{j}\right\}_{j=1}^{\infty}$ such that $I_{\varphi}(\nu_{s}|_{A_{j}})<\infty$, $\nu_{s}(A_{j})<\infty$, $A_{j}\subseteq X\setminus G$ for every $j\in\mathbb{N}$ and $\nu_{s}=\sum_{j=1}^{\infty}\nu_{s}|_{A_{j}}$.

We define the following decomposition of $\tau_{R}$ and $\nu_{R}$ to obtain the desired decomposition of the statement:

[TABLE]

and

[TABLE]

Then

[TABLE]

for every $i,j\in\mathbb{N}$. Let $i,j,k,l\in\mathbb{N}$ such that $E_{i,j}\neq E_{k,l}$. Then $E_{i,j}$ and $E_{k,l}$ are disjoint compact sets and let $r=\mathrm{dist}(E_{i,j},E_{k,l})$. Then

[TABLE]

[TABLE]

[TABLE]

Finally if $j,k,l\in\mathbb{N}$ then $A_{j}$ and $E_{k,l}$ are disjoint compact sets and let $r=\mathrm{dist}(A_{j},E_{k,l})$. Then

[TABLE]

[TABLE]

Hence decompositions of $\tau_{R}$ and $\nu_{R}$ in (9.11) and (9.12) satisfy the statement. ∎

Theorem 9.4**.**

Assume that (1.17), (1.19) and (1.22) hold and there exists $0<\delta<1$ such that (1.16), (1.23) and (1.18) hold. Let either $\nu$ and $\tau$ be finite Borel measures or $X$ be locally compact and $\nu$ and $\tau$ be locally finite Borel measures. Assume that $\nu(X\setminus X_{0})=0$ and $\tau(X\setminus X_{0})=0$. Assume that the conditional measure $\mathcal{C}(\nu)$ of $\nu$ and $\mathcal{C}(\tau)$ of $\tau$ on $B$ exist with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$. Assume that if $A\subseteq X$ is a Borel set such that $\nu(A)<\infty$ and $I_{\varphi}(\nu|_{A})<\infty$ then $\mathcal{C}_{k}(\nu)(A)$ converges to $\mathcal{C}(\nu)(A)$ in $\mathcal{L}^{2}$ and if $A\subseteq X$ is a Borel set such that $\tau(A)<\infty$ and $I_{\varphi}(\tau|_{A})<\infty$ then $\mathcal{C}_{k}(\tau)(A)$ converges to $\mathcal{C}(\tau)(A)$ in $\mathcal{L}^{2}$. Let $f:X\times X\longrightarrow\mathbb{R}$ be a nonnegative Borel function. Then

[TABLE]

[TABLE]

Proof.

By Lemma 9.3 we can decompose $\nu_{R}$ and $\tau_{R}$ into sum of finite measures such that $\nu_{R}=\sum_{i=1}^{\infty}\nu_{i}$, $\tau_{R}=\sum_{i=1}^{\infty}\tau_{i}$ and $I_{\varphi}(\nu_{i}+\tau_{i})<\infty$ for every $i,j\in\mathbb{N}$. By Theorem 9.2

[TABLE]

[TABLE]

for every $i,j\in\mathbb{N}$. By summing over all $i,j\in\mathbb{N}$ the statement follows by Property vii.), viii.) and ix.) in Definition 1.11. ∎

Proof of Theorem 1.21. By Remark 1.20 $\intop_{X}\intop_{X}\varphi(x,y)\left|f(x,y)\right|\mathrm{d}\nu(x)\mathrm{d}\tau(y)<\infty$ implies that $\intop F(x,y)\left|f(x,y)\right|\mathrm{d}\nu_{R}(x)\mathrm{d}\tau_{R}(y)<\infty$.

The conditional measure $\mathcal{C}(\nu)$ of $\nu$ and $\mathcal{C}(\tau)$ of $\tau$ on $B$ exist with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ by Theorem 8.1. The conditions of Theorem 9.4 are satisfied by Theorem 8.1. The statement follows by applying Theorem 9.4 to $f^{+}$ and $f^{-}$.$\hfill\square$

10 Conditional measure on an increasing

union

Our aim in this section is to establish the extension of the conditional measure with respect to an increasing union of $\mathcal{Q}_{k}^{i}$.

For every $i\in\mathbb{N}$ let $\mathcal{Q}_{k}^{i}$ be a sequence of countable families of Borel subsets of $X$ for $k\geq n_{i}$, for some $n_{i}\in\mathbb{N}$, such that $Q\cap S=\emptyset$ for $Q,S\in\mathcal{Q}_{k}^{i}$, for all $k\in\mathbb{N}$. Assume that (1.10), (1.11) and (1.13) hold. Assume further that if $i<j$ and $k\geq\max\{n_{i},n_{j}\}$ then

[TABLE]

See Example 1.6. Let $X_{0}^{i}=\bigcap_{k=n_{i}}^{\infty}(\bigcup_{Q\in\mathcal{Q}_{k}^{i}}Q)$ and let $X_{0}^{\infty}=\bigcup_{i=1}^{\infty}X_{0}^{i}$, note that it is an increasing union. Let either $\nu$ be a finite Borel measure on $X$ or $X$ be locally compact and $\nu$ be a locally finite Borel measure on $X$. Assume that

[TABLE]

and that the conditional measure $\mathcal{C}^{i}(\nu|_{X_{0}^{i}})$ of $\nu|_{X_{0}^{i}}$ on $B$ exists with respect to $\mathcal{Q}_{k}^{i}$ ($k\geq n_{i}$) with regularity kernel $\varphi$. Let

[TABLE]

with the convention that $X_{0}^{0}=\emptyset$.

Proposition 10.1**.**

Let $X_{0}^{i}$, $X_{0}^{\infty}$, $\mathcal{Q}_{k}^{i}$, $\nu$, $\mathcal{C}^{i}(\nu|_{X_{0}^{i}})$ and $\mu^{\nu}$ be as above. Then the following hold:

1.) $E(\intop_{X}f(x)\mathrm{d}\mu^{\nu}(x))=\intop_{X}f(x)\mathrm{d}\nu_{R}(x)$ for every $f:X\longrightarrow\mathbb{R}$ Borel measurable function such that $\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)<\infty$,

2.) $E(\mu^{\nu}(A))=\nu_{R}(A)\leq\nu(A)$ for every Borel set $A\subseteq X$ with $\nu(A)<\infty$,

3.) $\mu^{\nu_{\bot}}=0$ almost surely,

4.) $\mu^{\nu}=\mu^{\nu_{R}}$ almost surely,

5.) if $f:X\longrightarrow\mathbb{R}$ is a nonnegative Borel function such that $\intop_{X}f(x)\mathrm{d}\nu(x)<\infty$ then $\mu^{f(x)\mathrm{d}\nu(x)}=f(x)\mathrm{d}\mu^{\nu}(x)$ almost surely,

6.) $\mathrm{supp}\mu^{\nu}\subseteq\mathrm{supp}\nu\cap B$ almost surely,

Proof.

Let $f:X\longrightarrow\mathbb{R}$ be a nonnegative Borel measurable function such that $\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)<\infty$, then by Property ii.) of Definition 1.11 we have that $E(\intop_{X}f(x)\mathrm{d}\mathcal{C}^{i}(\nu|_{X_{0}^{i}\setminus X_{0}^{i-1}})(x))=\intop_{X}f(x)\mathrm{d}\nu_{R}|_{X_{0}^{i}\setminus X_{0}^{i-1}}(x)$ for every $i\in\mathbb{N}$. Then summing over $i\in\mathbb{N}$ and by Fubini‘s theorem we have that $E(\intop_{X}f(x)\mathrm{d}\mu^{\nu}(x))=\intop_{X}f(x)\mathrm{d}\nu_{R}(x)$, i.e. 1.) holds for nonnegative $f$. This implies 1.) in the general case since we know it for $f^{+}$ and $f^{-}$. Property 2.) is a spacial case of property 1.).

Property 3.), 4.), 5.) and 6.) follows by the fact that the analogous properties of the conditional measure in Definition 1.11 hold for the summands $\mathcal{C}^{i}(\nu|_{X_{0}^{i}\setminus X_{0}^{i-1}})$ (for Property 6.) note that $B$ is almost surely closed). ∎

Let $X_{0}^{i}$, $X_{0}^{\infty}$, $\mathcal{Q}_{k}^{i}$, $\nu$, $\mathcal{C}^{i}(\nu|_{X_{0}^{i}})$ and $\mu^{\nu}$ be as above. Let $\mathcal{Q}_{k}$ be a sequence of countable families of Borel subsets of $X$ such that $Q\cap S=\emptyset$ for $Q,S\in\mathcal{Q}_{k}$, for all $k\in\mathbb{N}$. Assume that (1.11), (1.13) hold and assume that

[TABLE]

for $k\geq n_{i}$ for every $i$. Let $\mathcal{C}_{k}(\nu)$ be as in (1.14).

Recall, that $X_{0}=\bigcap_{k=0}^{\infty}(\bigcup_{Q\in\mathcal{Q}_{k}}Q)$. Note that, if

[TABLE]

and $\nu(X\setminus X_{0})=0$ then (10.2) holds.

By (10.4), Property ix.) of Definition 1.11 and (10.1) it follows that

[TABLE]

Proposition 10.2**.**

Let $X_{0}^{i}$, $X_{0}^{\infty}$, $\mathcal{Q}_{k}^{i}$, $\mathcal{Q}_{k}$ $\nu$, $\mu^{\nu}$ and $\mathcal{C}_{k}(\nu)$ be as above, (note that we assume (10.2)). Then $\intop_{X}f(x)\mathrm{d}\mathcal{C}_{k}(\nu)(x)$ converges to $\int f(x)\mathrm{d}\mu^{\nu}(x)$ in probability for every Borel measurable function $f:X\longrightarrow\mathbb{R}$ such that $\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)<\infty$.

Proof.

Let $f$ be as in the statement. Let $\varepsilon>0$ be fixed. Since $\intop_{X}\left|f(x)\right|\mathrm{d}\nu(x)<\infty$ and we assume (10.2), we can find $i\in\mathbb{N}$ such that $\intop_{X\setminus X_{0}^{i}}\left|f(x)\right|\mathrm{d}\nu(x)<\varepsilon$. By (1.15) and (10.6) it follows that

[TABLE]

Since the conditional measure $\mathcal{C}^{j}(\nu|_{X_{0}^{j}})$ of $\nu|_{X_{0}^{j}}$ on $B$ exists with respect to $\mathcal{Q}_{k}^{j}$ ($k\geq n_{i}$) with regularity kernel $\varphi$ it follows by Property ii.), x.) and x.)* of Definition 1.11 that

[TABLE]

Thus by (10.3) and (10.6) it follows that

[TABLE]

[TABLE]

We have that $\intop_{X}f(x)\mathrm{d}\mathcal{C}_{k}^{i}(\nu|_{X_{0}^{i}})(x)$ converges to $\intop_{X}f(x)\mathrm{d}\mathcal{C}^{i}(\nu|_{X_{0}^{i}})(x)$ in probability by Property ii.) and iii.) of Definition 1.11. Thus using (10.7) and (10.8) it follows that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

It follows that

[TABLE]

because $\intop_{X}f(x)\mathrm{d}\mathcal{C}_{k}^{i}(\nu|_{X_{0}^{i}})(x)$ converges to $\intop_{X}f(x)\mathrm{d}\mathcal{C}^{i}(\nu|_{X_{0}^{i}})(x)$ in probability. Since $\varepsilon>0$ can be arbitrary the statement follows. ∎

Theorem 10.3**.**

Let $X_{0}^{i}$, $X_{0}^{\infty}$, $\mathcal{Q}_{k}^{i}$, $\mathcal{Q}_{k}$ $\nu$, $\mu^{\nu}$ and $\mathcal{C}_{k}(\nu)$ be as above. Then the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$, and $\mathcal{C}(\nu)=\mu^{\nu}$ almost surely.

Proof.

To show the existence of the conditional measure we wish to apply Theorem 1.25. Hence we need to show that $\mathcal{C}_{k}(\nu)(D)$ converges in $\mathcal{L}^{1}$ for every compact set $D\subseteq X_{0}$ with $I_{\varphi}(\nu|_{D})<\infty$ and $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability for every compact set $D\subseteq X$.

Let $D\subseteq X_{0}$ be a compact set such that $I_{\varphi}(\nu|_{D})<\infty$ which implies that $\nu_{R}(D)=\nu(D)$. Then $\mathcal{C}_{k}(\nu)(D)$ converges to $\mu^{\nu}(D)$ in probability by Proposition 10.2. By Property 2.) of Theorem 10.1 it follows that $E(\mu^{\nu}(D))=\nu_{R}(D)=\nu(D)$. It follows from (1.15) that $E(\mathcal{C}_{k}(\nu)(D))=\nu(D)$. Thus $\mathcal{C}_{k}(\nu)(D)$ converges to $\mu^{\nu}(D)$ in $\mathcal{L}^{1}$ by Lemma 2.10.

Let now $D\subseteq X$ be an arbitrary compact subset. Then $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to $\mu^{\nu_{\perp}}(D)$ in probability by Proposition 10.2. On the other hand by Property 3.) of Theorem 10.1 it follows that $\mu^{\nu_{\perp}}(D)=0$ almost surely. Thus $\mathcal{C}_{k}(\nu_{\perp})(D)$ converges to [math] in probability.

So we can conclude by the application of Theorem 1.16 that the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exist with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$. It remains to show that $\mu^{\nu}=\mathcal{C}(\nu)$ almost surely.

Assume first that $\nu$ is a finite Borel measure. Then $\mathcal{C}_{k}(\nu)(G)$ converges to $\mathcal{C}(\nu)(G)$ in probability for every open set $G\subseteq X$ by Property *ii.) *and iv.) of Definition 1.11. On the other hand, $\mathcal{C}_{k}(\nu)(G)$ converges to $\mu^{\nu}(G)$ in probability for every open set $G\subseteq X$ by Proposition 10.2. Hence $\mu^{\nu}=\mathcal{C}(\nu)$ almost surely by Lemma 3.25.

Now let $\nu$ be a locally finite Borel measure and let $X$ be locally compact. Then $\mathcal{C}_{k}(\nu)$ vaguely converges to $\mathcal{C}(\nu)$ in probability by Property i.) *of Definition 1.11. On the other hand, $\mathcal{C}_{k}(\nu)$ vaguely converges to $\mu^{\nu}$ in probability by Proposition 10.2. Since the limit is unique by Proposition 3.33 it follows that $\mu^{\nu}=\mathcal{C}(\nu)$ almost surely. ∎

Let $\overline{F^{i}}(x,y)$ and $\underline{F^{i}}(x,y)$ be defined as in Definition 1.19 for the sequence $\mathcal{Q}_{k}^{i}$ in place of $\mathcal{Q}_{k}$. If $x\notin X_{0}^{i}$ or $y\notin X_{0}^{i}$ then $\underline{F^{i}}(x,y)=\overline{F^{i}}(x,y)=0$. If $x,y\in X_{0}^{i}$ and $i<j$ then $\underline{F^{i}}(x,y)=\underline{F^{j}}(x,y)$ and $\overline{F^{i}}(x,y)=\overline{F^{j}}(x,y)$ by (10.1). Thus

[TABLE]

and

[TABLE]

are well-defined on $X\times X$. It is easy to see, by (10.4), that $\underline{F^{\infty}}(x,y)=\underline{F}(x,y)$ and $\overline{F^{\infty}}(x,y)=\overline{F}(x,y)$ for $(x,y)\in X_{0}^{\infty}\times X_{0}^{\infty}$. In particular, if (10.5) holds then $\underline{F^{\infty}}(x,y)=\underline{F}(x,y)$ and $\overline{F^{\infty}}(x,y)=\overline{F}(x,y)$ for every $(x,y)\in X\times X$ where $\underline{F}$ and $\overline{F}$ are defined in (1.19).

Theorem 10.4**.**

Let $\mathcal{Q}_{k}$ be a sequence of countable families of Borel subsets of $X$ such that $Q\cap S=\emptyset$ for $Q,S\in\mathcal{Q}_{k}$, for all $k\in\mathbb{N}$. Assume that if $C_{\varphi}(D)=0$ for some compact set $D\subseteq X_{0}$ then $B\cap D=\emptyset$ almost surely. For every $i\in\mathbb{N}$ let $\mathcal{Q}_{k}^{i}$ be a sequence of countable families of Borel subsets of $X$ for $k\geq n_{i}$, for some $n_{i}\in\mathbb{N}$, such that $Q\cap S=\emptyset$ for $Q,S\in\mathcal{Q}_{k}^{i}$, for all $k\in\mathbb{N}$. Assume that (1.17), (1.19), (1.22), (1.16), (1.23) and (1.18) hold for sufficient constants $0<\delta^{i}<1$, $M_{\delta}^{i}<\infty$, $0<M^{i}<\infty$, $0<a^{i}<\infty$, $0<c^{i}<\infty$, $c_{2}^{i}<\infty$, $c_{3}^{i}<\infty$ depending on $i$. Assume further that if $i<j$ and $k\geq\max\{n_{i},n_{j}\}$ then $\mathcal{Q}_{k}^{i}\subseteq\mathcal{Q}_{k}^{j}.$ Assume that (1.11), (1.13) hold and assume that $\mathcal{Q}_{k}^{i}\subseteq\mathcal{Q}_{k}$ for $k\geq n_{i}$ for every $i$. Let $X_{0}^{i}=\bigcap_{k=n_{i}}^{\infty}(\bigcup_{Q\in\mathcal{Q}_{k}^{i}}Q)$ and $X_{0}^{\infty}=\bigcup_{i=1}^{\infty}X_{0}^{i}$. Assume that $F(x,y)=\underline{F}(x,y)=\overline{F}(x,y)$ for every $x,y\in X$. Let either $\nu$ and $\tau$ be finite Borel measures on $X$ or $X$ be locally compact and $\nu$ and $\tau$ be locally finite Borel measures on $X$. Assume that $\nu(X\setminus X_{0}^{\infty})=0$ and $\tau(X\setminus X_{0}^{\infty})=0$. Then the conditional measure $\mathcal{C}(\nu)$ of $\nu$ and $\mathcal{C}(\tau)$ of $\tau$ on $B$ exist with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ and

[TABLE]

for every $f:X\times X\longrightarrow\mathbb{R}$ Borel function with $\intop\intop F(x,y)\left|f(x,y)\right|\mathrm{d}\nu_{R}(x)\mathrm{d}\tau_{R}(y)<\infty$.

Proof.

We have that $\overline{F^{\infty}}(x,y)=\overline{F^{i}}(x,y)$ and $\underline{F^{\infty}}(x,y)=\underline{F^{i}}(x,y)$ for $x,y\in X_{0}^{i}$ and that $F(x,y)=\underline{F^{\infty}}(x,y)=\overline{F^{\infty}}(x,y)$ for $x,y\in X_{0}^{\infty}$. It follows by Theorem 8.1 that the conditional measure $\mathcal{C}^{i}(\nu|_{X_{0}^{i}})$ of $\nu|_{X_{0}^{i}}$ on $B$ exists with respect to $\mathcal{Q}_{k}^{i}$ ($k\geq n_{i}$) with regularity kernel $\varphi$ for every $i$. Thus it follows by Theorem 10.3 that the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$. Similarly, the conditional measure $\mathcal{C}(\tau)$ of $\tau$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$.

Let $f:X\times X\longrightarrow\mathbb{R}$ be a Borel function with $\intop\intop F(x,y)\left|f(x,y)\right|\mathrm{d}\nu(x)\mathrm{d}\tau(y)<\infty$. By Property *ix.) *of Definition 1.11 it follows that $\mathcal{C}(\nu)=\sum_{i=1}^{\infty}\mathcal{C}(\nu|_{X_{0}^{i}\setminus X_{0}^{i-1}})$ and $\mathcal{C}(\tau)=\sum_{i=1}^{\infty}\mathcal{C}(\tau|_{X_{0}^{i}\setminus X_{0}^{i-1}})$. Let $i,j\in\mathbb{N}$, $i\leq j$. By (10.1) we have that $\mathcal{C}_{k}^{i}(\nu|_{X_{0}^{i}\setminus X_{0}^{i-1}})=\mathcal{C}_{k}^{j}(\nu|_{X_{0}^{i}\setminus X_{0}^{i-1}})$ and so $\mathcal{C}^{i}(\nu|_{X_{0}^{i}\setminus X_{0}^{i-1}})=\mathcal{C}^{j}(\nu|_{X_{0}^{i}\setminus X_{0}^{i-1}})$ almost surely. By applying Theorem 1.21 to the measures $\nu|_{X_{0}^{i}\setminus X_{0}^{i-1}}$ and $\tau|_{X_{0}^{j}\setminus X_{0}^{j-1}}$ with respect to the sequence $\mathcal{Q}_{k}^{j}$ it follows that

[TABLE]

[TABLE]

Similarly we can show that (10.9) holds when $i>j$. By (10.4) we have that $\mathcal{C}_{k}(\nu|_{X_{0}^{i}\setminus X_{0}^{i-1}})=\mathcal{C}_{k}^{i}(\nu|_{X_{0}^{i}\setminus X_{0}^{i-1}})$ and so $\mathcal{C}(\nu|_{X_{0}^{i}\setminus X_{0}^{i-1}})=\mathcal{C}^{i}(\nu|_{X_{0}^{i}\setminus X_{0}^{i-1}})$ and also $\mathcal{C}(\tau|_{X_{0}^{j}\setminus X_{0}^{j-1}})=\mathcal{C}^{i}(\tau|_{X_{0}^{j}\setminus X_{0}^{j-1}})$. Then summing (10.9) over all $i,j\in\mathbb{N}$ and by Fubini‘s theorem we can conclude that

[TABLE]

∎

11 Probability of non-extinction

In this section we estimate what is the probability that the conditional measure $\mathcal{C}(\nu)$ has positive total mass. Among other estimates we show Theorem 1.28. The upper bound in Theorem 1.28 is Corollary 11.7 and the lower bound follows from Corollary 11.5, Remark 1.20 and the double integration formula (1.24).

Let $K:X\times X\longrightarrow\mathbb{R}$ be a nonnegative Borel function. Recall the definition of the $K$-energy of a measure (1.20) and the $K$-capacity of a set (1.21).

Definition 11.1**.**

The $K$-capacity of a Borel measure $\nu$ is

[TABLE]

Definition 11.2**.**

The upper $K$-capacity of a Borel measure $\nu$ is

[TABLE]

It easily follows from the definitions that $C_{K}(\nu)\leq\overline{C_{K}}(\nu)$. In the next theorem $\mathcal{C}(\tau)$ denotes an abstract random measure that depends on $\tau$ and satisfies certain properties. Of course, then we apply the theorem to the conditional measure which satisfies the properties.

Theorem 11.3**.**

Let $\nu$ be a finite Borel measure and $K:X\times X\longrightarrow\mathbb{R}$ be a nonnegative Borel function such that

[TABLE]

whenever $I_{K}(\tau)<\infty$ for a finite Borel measure $\tau$ that satisfies $\tau\ll\nu$. Assume that $\mathcal{C}(\tau)$ is a random finite Borel measure on $X$ for every finite Borel measure $\tau$ that satisfies $\tau\ll\nu$ such that the following hold

1.) if $\nu_{i}$ are finite Borel measures and $\gamma_{i}\in\mathbb{R}$ are nonnegative such that $\sum_{i=1}^{\infty}\gamma_{i}\cdot\nu_{i}\leq\nu$ then $\sum_{i=1}^{\infty}\gamma_{i}\cdot\mathcal{C}(\nu_{i})\leq\mathcal{C}(\nu)$ almost surely,

2.) $E(\mathcal{C}(\tau)(X))\geq\tau_{R}(X)=\tau_{\varphi R}(X)$,

3.) $E\left(\mathcal{C}(\tau)(X)^{2}\right)\leq I_{K}(\tau)$.

Then

[TABLE]

Proof.

Let $\tau\ll\nu$ with $\tau(X)=1$ and $I_{K}(\tau)<\infty$ (if there is no such $\tau$ then the proof is trivial). By 2.) and (11.1) it follows that

[TABLE]

Let $A_{i}=\left\{x:i-1\leq\frac{\mathrm{d}\tau}{\mathrm{d}\nu}(x)<i\right\}$ for every $i\in\mathbb{N}$. Then $\nu=\sum_{i=1}^{\infty}\nu|_{A_{i}}$ and $\tau=\sum_{i=1}^{\infty}\tau|_{A_{i}}$. If $D\subseteq A_{i}$ is a Borel set then

[TABLE]

Hence $\sum_{i=1}^{\infty}\frac{1}{i}\mathcal{C}(\tau|_{A_{i}})\leq\mathcal{C}(\nu)$ almost surely by 1.). It follows that $\mathcal{C}(\nu)(X)=0$ implies $\mathcal{C}(\tau)(X)=0$. Thus

[TABLE]

Using the second moment method [19, Lemma 3.23], (11.2) and 3.) it follows that

[TABLE]

Taking the supremum over all $\tau$ it follows by (11.3) that $P(\mathcal{C}(\nu)(X)>0)\geq C_{K}(\nu)$. ∎

Remark 11.4*.*

Assume that $K:X\times X\longrightarrow\mathbb{R}$ is a nonnegative Borel function and $\nu\left(X\setminus\cup_{i=1}^{\infty}A_{i}\right)=0$ for some Borel sets $A_{i}$ such that there exists $0<a_{i}<\infty$ such that

[TABLE]

for every $x,y\in A_{i}$. Then $\tau_{\varphi\bot}(A_{i})=0$ whenever $I_{K}(\tau|_{A_{i}})<\infty$ for some finite Borel measure $\tau$ that satisfies $\tau\ll\nu$. If $I_{K}(\tau)<\infty$ for some $\tau$ then $I_{K}(\tau|_{A_{i}})\leq I_{K}(\tau)<\infty$ for every $i\in\mathbb{N}$. Hence $\tau_{\varphi\bot}(A_{i})=0$ for every $i\in\mathbb{N}$ and so (11.1) holds.

Corollary 11.5**.**

Let $\nu$ be a finite Borel measure. Assume that the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ and there exists $0<c<\infty$ such that

[TABLE]

for every finite Borel measure $\tau$ that satisfies $\tau\ll\nu$. Then

[TABLE]

Proof.

We show that the conditions of Theorem 11.3 hold for $K=c\cdot\varphi$ so we can conclude the statement from Theorem 11.3. Clearly (11.1) holds. Condition 1.) of Theorem 11.3 holds by Property ix.) and *x.) *of Definition 1.11. Condition 2.) of Theorem 11.3 holds by Property *v.) *of Definition 1.11. Condition 3.) of Theorem 11.3 holds by (11.5). ∎

Again, in the next theorem $\mathcal{C}(\nu|_{A})$ denotes an abstract random measure that depends on $A\subseteq X$ and satisfies certain properties and then we apply the theorem to the conditional measure which satisfies the properties.

Theorem 11.6**.**

Let $K:X\times X\longrightarrow\mathbb{R}$ be a nonnegative Borel function and let $\nu$ be a finite Borel measure such that $\nu(X\setminus X_{0})=0$ for some Borel set $X_{0}\subseteq X$. Let $B$ be a random closed set, assume that there exists $b>0$ such that

[TABLE]

for every compact set $D\subseteq X_{0}$. Assume that $\mathcal{C}(\nu|_{A})$ is a random finite Borel measure for every Borel set $A\subseteq X$ such that the following hold

1.) if $\left\{H_{i}\right\}_{i=1}^{\infty}$ is a sequence of disjoint Borel subsets of $X$ such that $\sum_{i=1}^{\infty}\nu|_{H_{i}}=\nu$ then $\mathcal{C}(\nu)=\sum_{i=1}^{\infty}\mathcal{C}(\nu|_{H_{i}})$ almost surely,

2.) for a compact set $D\subseteq X_{0}$ conditional on the event $D\cap B=\emptyset$ we have that $\mathcal{C}(\nu|_{D})(X)=0$ almost surely.

Then

[TABLE]

Proof.

Let $A_{n}\subseteq X_{0}$ be a sequence of Borel sets such that $\nu(X\setminus A_{n})=0$ and $C_{K}(A_{n})\leq\overline{C_{K}}(\nu)+1/n$. Then for $A:=\cap_{n=1}^{\infty}A_{n}$ we have that $\nu(X\setminus A)=0$ and $C_{K}(A)=\overline{C_{K}}(\nu)$. Let $D_{n}\subseteq A\subseteq X_{0}$ be an increasing sequence of compact sets such that $\nu(X\setminus D_{n})=\nu(A\setminus D_{n})<1/n$ (we can find such sequence by inner regularity). Conditional on $D_{n}\cap B=\emptyset$ we have that $\mathcal{C}(\nu|_{D_{n}})(X)=0$ almost surely by* 2.)*. Hence by (11.6)

[TABLE]

[TABLE]

Let $H_{1}=D_{1}$ and $H_{n}=D_{n}\setminus D_{n-1}$ for $n\geq 2$. Then $\nu|_{D_{n}}=\sum_{i=1}^{n}\nu|_{H_{i}}$ and $\nu=\sum_{i=1}^{\infty}\nu|_{H_{i}}$ since $\nu(X\setminus D_{n})=\nu(A\setminus D_{n})<1/n$. Hence $\mathcal{C}(\nu|_{D_{n}})=\sum_{i=1}^{n}\mathcal{C}(\nu|_{H_{i}})$ and $\mathcal{C}(\nu)=\sum_{i=1}^{\infty}\mathcal{C}(\nu|_{H_{i}})$ by 1.). Thus

[TABLE]

∎

Corollary 11.7**.**

Let $\nu$ be a finite Borel measure such that $\nu(X\setminus X_{0})=0$ and assume that the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$. Assume that $P(D\cap B\neq\emptyset)\leq b\cdot C_{\varphi}(D)$ for every compact set $D\subseteq X_{0}$. Then $P(\mathcal{C}(\nu)(X)>0)\leq b\cdot\overline{C_{\varphi}}(\nu)$.

Proof.

We show that conditions 1.)-2.) of Theorem 11.6 hold for $K=\varphi$. Condition 1.) holds by Property ix.) of Definition 1.11. For a compact set $D\subseteq X_{0}$ conditional on $D\cap B=\emptyset$ we have that $\mathcal{C}(\nu|_{D})(X)=0$ by Lemma 5.1, hence condition 2.) holds. ∎

Remark 11.8*.*

Let $X$ be locally compact and $\nu$ be a locally finite Borel measure on $X$. Then the conclusion of Theorem 11.3, Corollary 11.5, Theorem 11.6 and Corollary 11.7 hold for $\nu$. The proofs are identical to the proofs of the corresponding results.

Theorem 11.9**.**

Let $\mathcal{Q}_{k}$, $\mathcal{Q}_{k}^{i}$ and $X_{0}^{\infty}$ be as in Theorem 10.4. Assume that if $C_{\varphi}(D)=0$ for some compact set $D\subseteq X_{0}$ then $B\cap D=\emptyset$ almost surely. Assume that $F(x,y)=\underline{F}(x,y)=\overline{F}(x,y)$ for every $x,y\in X$ and there exists $b>0$ such that

[TABLE]

for every compact set $D\subseteq X_{0}$. Assume that $X_{0}=\cup_{i=1}^{\infty}A_{i}$ such that (11.4) holds. Let either $\nu$ be a finite Borel measure on $X$ or $X$ be locally compact and $\nu$ be a locally finite Borel measure on $X$. Assume that $\nu(X\setminus X_{0}^{\infty})=0$. Then the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ and

[TABLE]

Proof.

Let $K(x,y)=F(x,y)$. Since (11.4) holds it follows that (11.1) holds by Remark 11.4. The conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi$ and Condition 3.) of Theorem 11.3 holds by Theorem 10.4. Conditions 1.)-2.) of Theorem 11.3 hold by Property v.), ix.), x.) and x.)* of Definition 1.11. Thus it follows from Theorem 11.3 and Remark 11.8 that $C_{F}(\nu)\leq P(\mathcal{C}(\nu)(X)>0)$.

To prove the other inequality, we need to check that the conditions of Theorem 11.6 are satisfied to conclude from Theorem 11.6 and Remark 11.8 that $P(\mathcal{C}(\nu)(X)>0)\leq b\cdot\overline{C_{F}}(\nu)$. The conditions of Theorem 11.6 can be checked similarly to the proof of Corollary 11.7. ∎

12 Conditional measure on the Brownian path

Throughout this section let $B$ be the Brownian path in $\mathbb{R}^{d}$ ($d\geq 3$), i.e. the range of a Brownian motion which is started at the origin unless stated otherwise. We prove Theorem 1.29 and Theorem 1.22 in this section.

12.1 Strong Markov property

Let $W(t)$ be a Brownian motion started at the origin. Usually the strong Markov property is stated for stopping times that are almost surely finite. We reformulate the strong Markov property in the case when we only assume that the stopping time $T$ is finite with positive probability. We show that $W(t+T)-W(T)$ is a Brownian motion conditional on the event $\{T<\infty\}$ and is independent of what happens with $W$ up till time $T$.

Let $\mathcal{F}^{0}(t)$ be the $\sigma$-algebra generated by the random variables $\{W(s):0\leq s\leq t\}$, let $\mathcal{F}^{+}(t)=\bigcap_{\varepsilon>0}\mathcal{F}^{0}(t+\varepsilon)$ and for a stopping time $T$ let

[TABLE]

Proposition 12.1**.**

Let $\left(W(t):t\geq 0\right)$ be a Brownian motion started at the origin and let $T$ be a stopping time with respect to the filtration $\left(\mathcal{F}^{+}(t):t\geq 0\right)$. Then

[TABLE]

defines a Brownian motion conditional on $\{T<\infty\}$ that is independent of $\mathcal{F}^{+}(T)$.

Proof.

Let $a_{n}\geq 0$ be a monotone increasing deterministic sequence such that $\lim_{n\rightarrow\infty}a_{n}=\infty$ and $P(T=a_{n})=0$ for every $n\in\mathbb{N}$. Then let $T_{N}=\min\{T,a_{N}\}$ be stopping times and $W_{N}(t)=W(t+T_{N})-W(T_{N})$ be random processes for every $N\in\mathbb{N}$. Then

[TABLE]

for every $t\geq 0$, i.e. $\{T_{N}<a_{n}\}\in\mathcal{F}^{+}(T)$. Thus it follows by the strong Markov property (see for example [19, Theorem 2.14]) for $T_{N}$ that

[TABLE]

for large enough $N$ that $P(T<a_{N})>0$ and for every $A\in\mathcal{A}$. Taking the limit as $N$ goes to $\infty$ it follows that $P(W_{T}\in A\mid T<\infty)=P(W\in A)$ thus $W_{T}$ is a Brownian motion.

For the proof of the independence let $H\in\mathcal{F}^{+}(T)$ and $A\in\mathcal{A}$. Then by the strong Markov property for $T_{N}$

[TABLE]

[TABLE]

[TABLE]

Taking the limit as $N$ goes to $\infty$ it follows that

[TABLE]

then dividing both sides by $P(T<\infty)$ it follows that $\{W_{T}\in A\}$ and $\{W\in H\}$ are independent conditional on $\{T<\infty\}$. ∎

In the following sections we use the strong Markov property in the form of Proposition 12.1 for stopping times that are the first hitting of a compact set, hence the stopping time is not necessarily finite.

12.2 Existence of the kernel $F(x,y)$

This section is dedicated to show that $F(x,y)=\underline{F}(x,y)=\overline{F}(x,y)$ when $B$ is the Brownian path.

Lemma 12.2**.**

Let $B$ be a Brownian path in $\mathbb{R}^{d}$ for $d\geq 3$. Let $x\in\mathbb{R}^{d}$, $x\neq 0$ and $r>0$ such that $\left\|x\right\|>r$. Then

[TABLE]

See [19, Corollary 3.19].

Proposition 12.3**.**

Let $B$ be a Brownian path in $\mathbb{R}^{d}$ for $d\geq 3$. Then for $x,y\in\mathbb{R}^{d}\setminus\{0\}$, $x\neq y$

[TABLE]

[TABLE]

Proof.

Let $W(t)$ be a Brownian motion in $\mathbb{R}^{d}$ for some $d\geq 3$, so $B=\left\{W(t):t\in[0,\infty)\right\}$. We denote by $P=P_{0}$ the probability measure that corresponds to the Brownian motion that is started at the origin and by $P_{x}$ the probability measure that corresponds to the Brownian motion that is started at $x\in\mathbb{R}^{d}$. Let

[TABLE]

for $x\in\mathbb{R}^{d}$ and $r>0$. Let $x,y\in\mathbb{R}^{d}$ and $R,r>0$ such that $r+R<\left\|x-y\right\|$, $R<\left\|x\right\|$ and $r<\left\|y\right\|$. If $z\in\partial B(x,R)$ then

[TABLE]

by Lemma 12.2, thus

[TABLE]

and similarly for $z\in\partial B(y,r)$

[TABLE]

Let $U=W(T_{x,R})$ and $V=W(T_{y,r})$ be the stopped Brownian motions. Then by Lemma 12.2, (12.2) and by the strong Markov property (Proposition 12.1)

[TABLE]

[TABLE]

and similarly

[TABLE]

[TABLE]

Hence

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Hence by Lemma 12.2 it follows that

[TABLE]

[TABLE]

By (12.5) and Lemma 12.2 it follows that

[TABLE]

[TABLE]

and similarly

[TABLE]

Thus by Lemma 12.2, (12.2), (12.3), (12.7), (12.8) and by the strong Markov property (Proposition 12.1)

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Hence by Lemma 12.2 it follows that

[TABLE]

So (12.1) holds by (12.6) and (12.9). ∎

Lemma 12.4**.**

Let $B$ be a Brownian path in $\mathbb{R}^{d}$ for $d\geq 3$, let $0\in A\subseteq\mathbb{R}^{d}$ be a compact set with $\mathrm{diam}(A)>0$. Then for $x\in\mathbb{R}^{d}\setminus\{0\}$ and $0<r<2^{-1}\left\|x\right\|/\mathrm{diam}(A)$

[TABLE]

[TABLE]

where $G(x,y)=c(d)\left\|x-y\right\|^{2-d}$ is the Green‘s function of the Brownian motion for some constant $c(d)>0$ (see Remark 1.25).

Proof.

See [19, Theorem 3.33], that $G(x,y)$ is the Green‘s function of the Brownian motion. By [19, Corollary 8.12] and [19, Theorem 8.27]

[TABLE]

[TABLE]

On the other, hand by the scaling invariance of capacity it follows that

[TABLE]

We have that $\left\|x\right\|^{-1}(1-r\cdot\mathrm{diam}(A)/\left\|x\right\|)\leq\left(\left\|x\right\|+r\cdot\mathrm{diam}(A)\right)^{-1}$ and $\left(\left\|x\right\|-r\cdot\mathrm{diam}(A)\right)^{-1}\leq\left\|x\right\|^{-1}(1+2r\cdot\mathrm{diam}(A)/\left\|x\right\|)$ because $0<r<2^{-1}\left\|x\right\|/\mathrm{diam}(A)$. Thus the statement follows from (12.10) and (12.11). ∎

Proposition 12.5**.**

Let $B$ be a Brownian path in $\mathbb{R}^{d}$ for $d\geq 3$ , let $0\in A\subseteq\mathbb{R}^{d}$ be a compact set such that $C_{d-2}(A)>0$. Let $x,y\in\mathbb{R}^{d}\setminus\{0\}$, $x\neq y$ and $x_{R},y_{r}\in\mathbb{R}^{d}$ for every $r,R>0$ be such that $\lim_{R\rightarrow 0}x_{R}=x$ and $\lim_{R\rightarrow 0}y_{R}=y$. Then

[TABLE]

[TABLE]

For $x\in\mathbb{R}^{d}\setminus\{0\}$ let $Q_{k}(x)$ be the dyadic cube $\left([\frac{i_{1}}{2^{k}},\frac{i_{1}+1}{2^{k}})\times\dots\times[\frac{i_{d}}{2^{k}},\frac{i_{d}+1}{2^{k}})\right)\setminus\{0\}$ for $i_{1},\dots,i_{d}\in\mathbb{Z}$ such that $x\in Q_{k}(x)$ and let $\mathcal{Q}_{k}=\left\{Q_{k}(x):x\in\mathbb{R}^{d}\right\}$.

Proposition 12.6**.**

Let $B$ be a Brownian path in $\mathbb{R}^{d}$ for $d\geq 3$. Then

[TABLE]

[TABLE]

The proof of Proposition 12.5 goes similarly to the proof of Proposition 12.3 replacing the use of Lemma 12.2 by Lemma 12.4.

Remark 12.7*.*

Proposition 12.6 is a special case of Proposition 12.5. Note that it is not an issue that the cubes are not compact because the hitting probability does not depend on whether the boundary of the cube is in the set or not. It can, for example, be deduced from Lemma 12.4 by approximating the cube by compact cubes from inside and outside.

Theorem 12.8**.**

Let $B$ be a Brownian path in $\mathbb{R}^{d}$ for $d\geq 3$. Let $\mathcal{Q}_{k}$ be as in Example 1.5 then

[TABLE]

for $x,y\in\mathbb{R}^{d}\setminus\{0\}$, $x\neq y$.

Theorem 12.8 is a reformulation of Proposition 12.6.

12.3 Conditions on $\mathcal{Q}_{k}^{i}$

For the rest of Section 12 let $\mathcal{Q}_{k}$ be as in Example 1.5, $\mathcal{Q}_{k}^{i}$ be as in Example 1.6 and let $\varphi(r)=r^{-(d-2)}$. In Section 12.3 we show that the assumptions of Section 1.1.2 hold for the sequence $\mathcal{Q}_{k}^{i}$ for sufficient constants.

We have that (1.16) holds for every $\delta>0$ for sufficient $c_{2}$ depending on $\delta$ and for $c_{3}=0$. We have that (1.17) holds. Clearly (1.10), (1.11), (1.13), (1.19) hold for both $\mathcal{Q}_{k}$ and $\mathcal{Q}_{k}^{i}$ for every $i\in\mathbb{N}$ with $M$=1. We have that $Q\cap S=\emptyset$ whenever $Q\neq S$ and $Q,S\in\mathcal{Q}_{k}$ or $Q,S\in\mathcal{Q}_{k}^{i}$. It is easy to see that (1.18) hold for $\mathcal{Q}_{k}^{i}$ for every $i\in\mathbb{N}$ for every $\delta>0$ for constant $M_{\delta}>0$ that only depends on $\delta$ and $d$.

It follows from Lemma 12.4, Remark 12.7 and the scaling invariance of capacity (12.11) that (1.22) holds for $\mathcal{Q}_{k}^{i}$ for some sufficient constant $a^{i}<\infty$.

Lemma 12.9**.**

Let $B$ be a Brownian path in $\mathbb{R}^{d}$ for $d\geq 3$ and let $i$ be a positive integer. Then there exist $0<\delta=\delta^{i}<1$ and $0<c=c^{i}<\infty$ such that (1.23) holds for $\mathcal{Q}_{k}^{i}$ $(k\in\mathbb{N},k\geq i)$ and $c$ and $\delta$ depends only on $i$ and $d$.

Proof.

Let $W(t)$, $P_{x}$ and $T_{x,r}$ be as in the proof of Proposition 12.3. Let $A=[0,1)^{d}.$ Let $N\in\mathbb{N}$ be large enough that

[TABLE]

Let $\delta\leq 1/(2\sqrt{d})$, $k,n\geq\max\{i,N\}$ and $x\in Q\in\mathcal{Q}_{k}^{i}$, $y\in S\in\mathcal{Q}_{n}^{i}$ such that $\max\left\{\mathrm{\mathrm{diam}}(Q),\mathrm{\mathrm{diam}}(S)\right\}<\delta\cdot\mathrm{dist}(Q,S)$. Let $r=2^{-k}$ and $R=2^{-n}$. Then

[TABLE]

for every $z\in\overline{Q}$. So

[TABLE]

for every $z\in\overline{Q}$ and. Thus by Lemma 12.4 and Remark 12.7

[TABLE]

for $a_{0}=C_{G}(A)\cdot 2^{d-2}$ for every $z\in\overline{Q}$.

By (12.12) it follows that

[TABLE]

because $x\in Q\in\mathcal{Q}_{k}^{i}$. Hence by Lemma 12.4

[TABLE]

Similarly

[TABLE]

Then similarly to (12.4) it follows from (12.14) and (12.13) that

[TABLE]

[TABLE]

for $b=C_{G}(A)2^{d-2}a_{0}$. Similarly we can show that

[TABLE]

Thus

[TABLE]

[TABLE]

[TABLE]

Hence it follows from (12.14) and (12.15) that

[TABLE]

for $c=2(\sqrt{d}2^{i})^{(d-2)}b(2^{d-2})^{2}/C_{G}(A)^{2}$.

So (1.23) holds if $k,n\geq\max\{i,N\}$ and $\delta\leq 1/(2\sqrt{d})$. We can choose $\delta>0$ to be small enough such that if either $k<\max\{i,N\}$ or $n<\max\{i,N\}$ then $\max\left\{\mathrm{diam}(Q),\mathrm{diam}(S)\right\}<\delta\cdot\mathrm{dist}(Q,S)$ does not hold for every pair of $Q\in\mathcal{Q}_{k}^{i}$ and $D\in\mathcal{Q}_{n}^{i}$. ∎

12.4 Existence

of the conditional measure on the Brownian path

We show Theorem 1.22 in this section.

Lemma 12.10**.**

If $A\subseteq\mathbb{R}^{d}\setminus\{0\}$ is a compact set and $C_{d-2}(A)=0$ then $P(B\cap A\neq\emptyset)=0$.

Lemma 12.10 is a direct corollary of Proposition 1.30

Proof of Theorem 1.22. In Section 12.3 we establish that the assumptions of Section 1.1.2 hold for sufficient constants $c_{2}^{i}<\infty$, $c_{3}=0$, $0<\delta^{i}<1$, $M_{\delta}^{i}<\infty$, $M=1$, $a^{i}<\infty$ and $0<c^{i}<\infty$ for the sequence $\mathcal{Q}_{k}^{i}$. We have that $\mathcal{Q}_{k}^{i}\subseteq\mathcal{Q}_{k}^{j}\subseteq\mathcal{Q}_{k}$ for $i\leq j\leq k$. It is easy to see that $X_{0}^{\infty}=\cup_{i\in\mathbb{N}}X_{0}^{i}=\mathbb{R}^{d}\setminus\{0\}$. In Theorem 12.8 we prove that

[TABLE]

Thus along with Lemma 12.10 the conditions of Theorem 10.4 are satisfied. Hence the statement follows from Theorem 10.4.$\hfill\square$

12.5 Probability of non-extinction

of the conditional measure on the Brownian path

Theorem 1.29 states an analogous result to Proposition 1.30 for measures.

Proof of Theorem 1.29. We wish to apply Theorem 11.9 for $K(x,y)=F(x,y)$ to conclude Theorem 1.29. In the proof of 1.22 we show that the conditions of Theorem 10.4 are satisfied. Obviously we can find a decomposition $\mathbb{R}^{d}\setminus\{0\}=\cup_{i=1}^{\infty}A_{i}$ such that (11.4) holds, namely $A_{i}=X_{0}^{i}$. So, along with Proposition 1.30, the conditions of Theorem 11.9 are satisfied. The explicit formula for $F(x,y)$ is given in Theorem 12.8.$\hfill\square$

13 Conditional measure of the Lebesgue measure on the Brownian path

and the occupation measure

In this Section we prove Theorem 1.24 that is the union of Theorem 13.18 and Theorem 13.19. The proof consists many steps that we sorted in three sections. In Section 13.1 we establish an asymptotic result on the number of cubes of side length $1$ that is intersected by the Brownian path. We prove this by the application of the ergodic theorem. The major part of the proof is to show the ergodicity of the invariant measure that we define. Then in Section 13.2 we use the scaling invariance of the Brownian motion to deduce an asymptotic result on the number of small cubes that is intersected by the Brownian path. Finally, in Section 13.3 we use the result on the number of small cubes that is intersected by the Brownian path and an approximation argument to finish the proof.

For a random variable $Y$ and a $\sigma$-algebra $\mathcal{F}\subseteq\mathcal{A}$ we denote the conditional expectation of $Y$ with respect to $\mathcal{F}$ by $E\left(Y\mid\mathcal{F}\right)$. For random variables $Y$ and $Z$ we write $E\left(Y\mid Z\right)$ for $E\left(Y\mid\mathcal{F}\right)$ where $\mathcal{F}$ is the $\sigma$-algebra generated by $Z$. In that case there exists a deterministic function $f$ such that $E\left(Y\mid Z\right)=f(Z)$ almost surely and we write $E\left(Y\mid Z=z\right)$ for $f(z)$. We write $E\left(Y\mid Z_{1},\dots,Z_{n}\right)$ when $\mathcal{F}$ is the $\sigma$-algebra generated by $Z_{1},\dots,Z_{n}$. In the rest of the paper we use many basic properties of the conditional expectation without reference. For an overview of the conditional expectation see for example [6, 8].

Throughout this section let $\mathcal{Q}_{k}$ be as in Example 1.5 and let

[TABLE]

for $k\in\mathbb{N}$. Recall that we denote the Lebesgue measure by $\lambda$.

13.1 Application of the ergodic theorem

Throughout this subsection let $B_{i}$ ($i\in\mathbb{Z}$) be an i.i.d. sequence of Brownian motions started at $0\in\mathbb{R}^{d}$ with domain $[0,1]$. Let $X$ be a random variable uniformly distributed on $[0,1)^{d}$ such that $X$ and $B_{i}$ ($i\in\mathbb{Z}$) are mutually independent. Let $W_{0}=W_{0}^{\omega}$ be the random function defined by

[TABLE]

and let $W(t)=X+W_{0}(t)$, i.e. $W$ is a two sided Brownian motion started at $X$ (it is due to the independent increments of the Brownian motion). Note, that $B_{i}(.)=\left(W(.+i)-W(i)\right)|_{[0,1]}$. Thus $W$ determines $X$ and the sequence $B_{i}$ and vice versa.

For a vector $v=(v_{1},\dots,v_{d})\in\mathbb{R}^{d}$ we denote by $\{v\}$ the equivalence class of $v$ in $\mathbb{R}^{d}/\mathbb{Z}^{d}$ and we denote by $\{v\}_{0}$ the element of the equivalence class of $v$ in $\mathbb{R}^{d}/\mathbb{Z}^{d}$ that is contained in $[0,1)^{d}$. For an equivalence class $w\in\mathbb{R}^{d}/\mathbb{Z}^{d}$ we also denote by $\{w\}_{0}$ the element of the equivalence class $w$ that is contained in $[0,1)^{d}$. Let $X_{n}=\{W(n)\}$.

Let $S$ be the right shift map on $C([0,1],d)^{\mathbb{Z}}\times(\mathbb{R}^{d}/\mathbb{Z}^{d})^{\mathbb{Z}}$, where $C(K,d)$ denotes the space of continuous functions from the compact set $K$ to $\mathbb{R}^{d}$ equipped with the supremum norm. So $S\left((f_{i})_{i=-\infty}^{\infty},(x_{i})_{i=-\infty}^{\infty}\right)=\left((f_{i+1})_{i=-\infty}^{\infty},(x_{i+1})_{i=-\infty}^{\infty}\right)$ for $\left((f_{i})_{i=-\infty}^{\infty},(x_{i})_{i=-\infty}^{\infty}\right)\in C([0,1],d)^{\mathbb{Z}}\times\left(\mathbb{R}^{d}/\mathbb{Z}^{d}\right)^{\mathbb{Z}}$. Let $R$ be the probability distribution of $\left((B_{i})_{i=-\infty}^{\infty},(X_{i})_{i=-\infty}^{\infty}\right)$. Then $R$ is a Borel measure. We show that $R$ is a shift invariant ergodic measure.

Lemma 13.1**.**

We have that $R$ is a shift invariant measure, i.e. $R(.)=R(S^{-1}(.))$.

Proof.

Since $X_{i}=\{W(i)\}$ and $B_{i}(.)=\left(W(.+i)-W(i)\right)|_{[0,1]}$ we need to show that the random functions $W(.)$ and $W(.-1)-\left\lfloor W(-1)\right\rfloor$ have the same distribution, where $\left\lfloor.\right\rfloor$ denotes the floor function in every coordinates. We have that

[TABLE]

for ever Borel set $A\subseteq\mathbb{R}^{d}/\mathbb{Z}^{d}$ and $H\subseteq C([0,1],d)$ because $X_{0}$ is uniformly distributed on $\mathbb{R}^{d}/\mathbb{Z}^{d}$ and is independent of $B_{-1}$. Hence

[TABLE]

[TABLE]

Since the collection of sets of the form $A\times H$ is a semi-ring generating the Borel $\sigma$-algebra it follows from Proposition 2.33 that $B_{-1}$ and $\left\{W(-1)\right\}=X_{0}-\left\{B_{-1}(1)\right\}$ are independent. Obviously we have that $B_{i}$ ($i\in\mathbb{Z}\setminus\{-1\}$) are mutually independent of $B_{-1}$ and $\left\{W(-1)\right\}=X_{0}-\left\{B_{-1}(1)\right\}$ hence $W_{0}(.-1)-W_{0}(-1)$ is a two sided Brownian motion started at [math] such that $\left\{W(-1)\right\}$ and $W_{0}(.-1)-W_{0}(-1)$ are independent. Hence $W(.)$ and $W(.-1)-\left\lfloor W(-1)\right\rfloor=\{W(-1)\}_{0}+W_{0}(.-1)-W_{0}(-1)$ have the same distribution. ∎

Lemma 13.2**.**

Let $Y$ be a random variable that takes values in $[0,1)$ almost surely, the distribution of $Y$ is absolutely continuous with respect to the Lebesgue measure and the density function $f$ is a bounded Riemann-integrable function. Then for every Borel set $A\subseteq[0,1)$

[TABLE]

where $U$ is a random variable uniformly distributed on $[0,1)$. Let $M<\infty$ be such that $0\leq f(x)\leq M$ for Lebesgue almost every $x\in\mathbb{R}$. Then the density function of $\{kY\}_{0}$ is bounded by $M$ for Lebesgue almost every $x\in\mathbb{R}$.

Proof.

The density function of $kY$ is $k^{-1}f(x/k)$ for Lebesgue almost every $x$ and so the density function of $\{kY\}_{0}$ is $k^{-1}\sum_{i=0}^{k-1}f((x+i)/k)$ for Lebesgue almost every $x\in[0,1)$ where we used the fact that $f(x)=0$ for $x\notin[0,1)$. For every $x\in[0,1)$ we have that $k^{-1}\sum_{i=0}^{k}f((x+i)/k)$ converges to $1$ since $f$ is Riemann integrable. The sequence $k^{-1}\sum_{i=0}^{k-1}f((x+i)/k)$ is uniformly bounded by $M$ for Lebesgue almost every $x\in\mathbb{R}$. Hence by the dominated convergence theorem $\lim_{k\rightarrow\infty}P\left(\{kY\}_{0}\in A\right)=P(U\in A)$ for every Borel set $A\subseteq[0,1)$. ∎

Lemma 13.3**.**

Let $Y$ be a normally distributed vector in $\mathbb{R}^{d}$ such that the coordinates are independent. Then for Borel sets $A_{1},\dots,A_{d}\subseteq\mathbb{R}/\mathbb{Z}$

[TABLE]

where $U$ is a random variable uniformly distributed on $\mathbb{R}^{d}/\mathbb{Z}^{d}$. Additionally, the density functions of $\{kY\}_{0}$ are uniformly bounded.

Proof.

We prove the statement for standard normally distributed vector $Y$ but the proof goes similarly for the general $Y$. Let $Y=(Y_{1},\dots,Y_{d})$ and $U=(U_{1},\dots,U_{d})$. The sequence of functions $\sum_{i=-n}^{n}e^{-(x+i)^{2}/2}$ locally uniformly converges to

[TABLE]

hence $f$ is a continuous function and so

[TABLE]

is a bounded Riemann-integrable function. Since $Y_{j}$ is standard normally distributed in $\mathbb{R}$ it follows that $\frac{1}{\sqrt{2\pi}}g$ is the density function of $\{Y_{j}\}_{0}$, thus

[TABLE]

and the density function of $\{kY_{j}\}_{0}$ is bounded by $\frac{1}{\sqrt{2\pi}}\sup_{x\in[0,1)}g(x)$ by Lemma 13.2. Since $Y_{j}$ are independent for $j=1,\dots,d$ it follows that

[TABLE]

[TABLE]

and the density function of $\{kY\}_{0}$ is uniformly bounded by $\left(\frac{1}{\sqrt{2\pi}}\sup_{x\in[0,1)}g(x)\right)^{d}$. ∎

We say that $C\subseteq\mathbb{R}^{d}/\mathbb{Z}^{d}$ is a box in $\mathbb{R}^{d}/\mathbb{Z}^{d}$ if there exists $0\leq a_{j}\leq b_{j}\leq 1$ ($j=1,\dots,d$) such that $C=\{A\}=\left\{\{a\}:a\in A\right\}$ for $A=[a_{1},b_{1})\times\dots\times[a_{d},b_{d})$.

Lemma 13.4**.**

We have that $R$ is mixing, i.e.

[TABLE]

for Borel sets $A_{1},A_{2}\subseteq C([0,1],d)^{\mathbb{Z}}\times\left(\mathbb{R}^{d}/\mathbb{Z}^{d}\right)^{\mathbb{Z}}$.

Proof.

Since $R$ is shift invariant by Lemma 13.1, due to [21, Theorem 1.17] it is enough to show that the statement holds for every $A_{1}$ and $A_{2}$ taken from a semi-ring of Borel sets that generates the Borel $\sigma$-algebra. Hence it is enough to show the statement for Borel sets of the form $(H_{i})_{i=-\infty}^{\infty}\times(C_{i})_{i=-\infty}^{\infty}$ where $H_{i}\subseteq C([0,1],d)$ is a Borel set, $C_{i}$ is a box in $\mathbb{R}^{d}/\mathbb{Z}^{d}$ for every $i\in\mathbb{Z}$ and there exists $n\in\mathbb{N}$ such that $H_{i}=C([0,1],d)$ and $C_{i}=\mathbb{R}^{d}/\mathbb{Z}^{d}$ for $i\in\mathbb{Z}\setminus[-n,n]$.

Let $A_{1}=(H_{i})_{i=-\infty}^{\infty}\times(C_{i})_{i=-\infty}^{\infty}$ and $A_{2}=(G_{i})_{i=-\infty}^{\infty}\times(D_{i})_{i=-\infty}^{\infty}$ where $H_{i},G_{i}\subseteq[0,1)^{d}$ are Borel sets, $C_{i}$, $D_{i}$ are boxes in $\mathbb{R}^{d}/\mathbb{Z}^{d}$ for every $i\in\mathbb{Z}$ and there exists $n\in\mathbb{N}$ such that $H_{i}=G_{i}=C([0,1],d)$ and $C_{i}=D_{i}=[0,1)^{d}$ for $i\in\mathbb{Z}\setminus[-n,n]$ (note, that without the loss of generality we can assume that $n$ is the same for both $A_{1}$ and $A_{2}$). Then

[TABLE]

Let $k>2n$. Then

[TABLE]

[TABLE]

because $X_{i-k}$ depends only on $\left(X_{-n-k},(B_{i})_{i=-n-k}^{n-k}\right)$. We have that

[TABLE]

where $X_{n-k}=\left\{W(n-k)\right\}$ depends only on $\left(X_{-n-k},(B_{i})_{i=-n-k}^{n-k}\right)$, $Z_{i}:=\left\{W(i)-W(-n)\right\}=\left\{W_{0}(i)-W_{0}(-n)\right\}$ depends only on $(B_{i})_{i=-n}^{n}$, $\left\{W(-n)-W(n-k)\right\}=\left\{W_{0}(-n)-W_{0}(n-k)\right\}$ is independent of $\left(X_{-n-k},(B_{i})_{i=-n}^{n},(B_{i})_{i=-n-k}^{n-k}\right)$, and $\left\{W(-n)-W(n-k)\right\}$ and $\left\{(k-2n)Y\right\}$ have the same distribution for a standard normally distributed vector $Y$ that is independent of $\left(X_{-n-k},(B_{i})_{i=-n}^{n},(B_{i})_{i=-n-k}^{n-k}\right)$. Thus

[TABLE]

[TABLE]

[TABLE]

and so it follows from (13.2) that

[TABLE]

[TABLE]

Let $\left(\widetilde{B_{i}},\widetilde{X_{i}}\right)_{i=-n}^{n}$ be a random variable with the same distribution as $\left(B_{i-k},X_{i-k}\right)_{i=-n}^{n}$ such that $\left(\widetilde{B_{i}},\widetilde{X_{i}}\right)_{i=-n}^{n}$ is independent of $\left(Y,X_{-n-k},(B_{i})_{i=-n}^{n},(B_{i})_{i=-n-k}^{n-k}\right)$. Then by the shift invariance (Lemma 13.1) $\left(\widetilde{B_{i}},\widetilde{X_{i}}\right)_{i=-n}^{n}$ and $\left(B_{i},X_{i}\right)_{i=-n}^{n}$ have the same distribution. Then

[TABLE]

and

[TABLE]

have the same distribution. Hence by (13.1) and (13.3) it follows that

[TABLE]

[TABLE]

It follows from Lemma 13.3 that

[TABLE]

[TABLE]

[TABLE]

almost surely where $U$ is a random variable that is independent of $\left(\widetilde{X_{-n}},(B_{i})_{i=-n}^{n},(\widetilde{B_{i}})_{i=-n}^{n}\right)$ and is uniformly distributed in $\mathbb{R}^{d}/\mathbb{Z}^{d}$. Thus by the dominated convergence theorem and (13.4) it follows that

[TABLE]

[TABLE]

Since $\left(U,(Z_{i})_{i=-n}^{n},(B_{i})_{i=-n}^{n}\right)$ is independent of $\left(\widetilde{X_{-n}},(\widetilde{B_{i}})_{i=-n}^{n}\right)$ it follows that

[TABLE]

[TABLE]

Thus by (13.5)

[TABLE]

[TABLE]

[TABLE]

[TABLE]

since $\prod_{i=-n}^{n}I_{\widetilde{B_{i}}\in G_{i}}I_{\widetilde{X_{i}}\in D_{i}}$ and $E\left(\prod_{i=-n}^{n}I_{B_{i}\in H_{i}}I_{U+Z_{i}\in C_{i}}\mid(B_{i})_{i=-n}^{n}\right)$ are independent and that $\left(\widetilde{B_{i}},\widetilde{X_{i}}\right)_{i=-n}^{n}$ and $\left(B_{i},X_{i}\right)_{i=-n}^{n}$ have the same distribution.

Similarly to the proof of Lemma 13.1 it can be shown that $U-\left\{W_{0}(-n)\right\}$ and $W_{0}$ are independent and $W=X+W_{0}$ and $\left\{U-\left\{W_{0}(-n)\right\}\right\}_{0}+W_{0}$ have the same distribution. Since $Z_{i}=\left\{W_{0}(i)\right\}-\left\{W_{0}(-n)\right\}$ it follows that

[TABLE]

and this finishes the proof combined with (13.6). ∎

Lemma 13.5**.**

Let

[TABLE]

i.e. the number of dyadic cubes of side length $1$ that is visited by $W$ in the time interval $[0,1)$ but never visited before. Let

[TABLE]

i.e. the number of dyadic cubes of side length $1$ that is visited by $W$ in the time interval $[0,1]$. Then $f$ and $g$ are $\mathcal{L}^{1}$ functions with respect to the probability measure $R$.

Proof.

It is easy to see that $f$ and $g$ are measurable and that $0\leq f\leq g$. Thus to prove the statement it is enough to show that

[TABLE]

Let $Y_{1}(t)$ be a Brownian motion in $\mathbb{R}$ started at [math] and let $x_{1}\in[0,1)$. Then

[TABLE]

[TABLE]

Note that $\mathcal{Q}_{0}^{*}$ consists dyadic intervals of the line in (13.7) because $Y_{1}$ is a Brownian motion in $\mathbb{R}$, however, later on $Y$ is a Brownian path in $\mathbb{R}^{d}$ and so $\mathcal{Q}_{0}^{*}$ consists dyadic cubes of $\mathbb{R}^{d}$. Let $Y(t)=(Y_{1}(t),\dots,Y_{d}(t))$ be a Brownian motion in $\mathbb{\mathbb{R}}^{d}$ started at [math] and let $(x_{1},\dots,x_{d})\in[0,1)^{d}$. Then by (13.7)

[TABLE]

Since $Y_{1},\dots,Y_{d}$ are mutually independent it follows that

[TABLE]

because $E\left(\max_{t\in[0,1]}\left|Y_{i}(t)\right|\right)<\infty$ by [19, Theorem 2.21]. ∎

Lemma 13.6**.**

There exist $0\leq\alpha\leq\beta<\infty$ such that

[TABLE]

$R$* almost surely and*

[TABLE]

$R$* almost surely.*

Proof.

Let $f$ and $g$ be as in Lemma 13.5. Then

[TABLE]

(note that $W(N)\notin\partial Q$ almost surely for every $N\in\mathbb{Z}$ and $Q\in\mathcal{Q}_{0}^{*}$) and

[TABLE]

By Lemma 13.5 the functions $f$ and $g$ are $\mathcal{L}^{1}$ functions with respect to the probability measure $R$, by Lemma 13.1 we have that $R$ is a shift invariant measure and by Lemma 13.4 we have that $R$ is an ergodic measure since mixing implies ergodicity. Hence by Birkhoff‘s ergodic theorem ([21, Theorem 1.14] and the Remark after the theorem) the statement follows for

[TABLE]

and

[TABLE]

∎

Proposition 13.7**.**

Let $0\leq\alpha\leq\beta<\infty$ be as in Lemma 13.6. Let $\widehat{W_{n}}(t)$ be two-sided Brownian motions in $\mathbb{R}^{d}$ started at [math] for every $n\in\mathbb{N}$. Let $Y_{n}$ be a random variable for every $n\in\mathbb{N}$, that is independent of $\widehat{W_{n}}$ (not necessarily i.i.d.), takes values in $\mathbb{R}^{d}$, the distributions of $Y_{n}$ are absolutely continuous with respect to the Lebesgue measure and the density functions of $\left\{Y_{n}\right\}_{0}$ are uniformly bounded. Let $\widetilde{W_{n}}=\widehat{W_{n}}+Y_{n}$. Then

[TABLE]

converges to $\alpha$ in probability as $N$ goes to $\infty$ and

[TABLE]

converges to $\beta$ in probability as $N$ goes to $\infty$.

Proof.

The quantity in (13.10) does not change if we replace $Y_{n}$ by $\left\{Y_{n}\right\}_{0}$, hence without the loss of generality we assume that $Y_{n}=\left\{Y_{n}\right\}_{0}$. Let $\varepsilon>0$ be fixed. Let

[TABLE]

then

[TABLE]

because $\widehat{W_{n}}$ and $W_{0}$ have the same distribution. Let $D_{n}(x)$ be the density function of $\left\{Y_{n}\right\}_{0}$ and let $M<\infty$ be the uniform bound that $D_{n}(x)\leq M$ for every $n\in\mathbb{N}$ and Lebesgue almost every $x\in[0,1)^{d}$. Then

[TABLE]

by (13.12). We have that $h_{N}$ converges to $\alpha$ almost surely by Lemma 13.6 and so in probability. Hence

[TABLE]

by (13.13). Similarly we can show that

[TABLE]

and hence

[TABLE]

i.e. $f_{n}$ converges to $\alpha$ in probability as $N$ goes to $\infty$.

The proof of

[TABLE]

for every $\varepsilon>0$ is similar to the proof of (13.14), we omit the details. ∎

13.2 Number of small cubes intersected by the Brownian path

We say that a number $a\in\mathbb{R}$ is a dyadic number if $a=n\cdot 2^{-k}$ for some $n\in\mathbb{Z}$, $k\in\mathbb{N}$. We say that an interval $J\subseteq\mathbb{R}$ is an interval with dyadic endpoints if there exist dyadic numbers $a<b$ such that $J$ is one of the following intervals $(a,b)$, $[a,b]$, $[a,b)$, $(a,b]$.

Throughout this subsection let $W_{0}(t)$ be a two-sided Brownian motion in $\mathbb{R}^{d}$ started at [math]. For a compact interval $I=[a,b]\subseteq\mathbb{R}$ with dyadic endpoints we define the following random variables:

[TABLE]

and

[TABLE]

where $N_{k}=(b-a)2^{2k}$ (note, that $N_{k}$ is an integer for large enough $k$ because $(b-a)$ is also a dyadic number).

Lemma 13.8**.**

Let $r>0$ be fixed. Then $\left(W_{0}(t):t\in\mathbb{R}\right)$ and $\left(r^{-1}W_{0}(r^{2}t):t\in\mathbb{R}\right)$ have the same distribution.

Lemma 13.8 is a folklore in the theory of Brownian motions, see for example [19, Lemma 1.7].

Lemma 13.9**.**

Let $0\leq\alpha\leq\beta<\infty$ be as in Lemma 13.6. Let $W_{0}(t)$ be a two-sided Brownian motion in $\mathbb{R}^{d}$ ($d\geq 3$) started at [math]. Then for every $0<a<b<\infty$ dyadic numbers and for $I=[a,b]$ we have that $f_{k}^{I}$ converges to $\alpha(b-a)$ in probability as $k$ goes to $\infty$ and $g_{k}^{I}$ converges to $\beta(b-a)$ in probability as $k$ goes to $\infty$.

Proof.

Let $0<a<b<\infty$ be fixed dyadic numbers. Assume that $k$ is large enough that $N_{k}$ is an integer. Let $\widetilde{W_{k}}(t)=2^{k}W_{0}(2^{-2k}t+a))$ for every $t\in\mathbb{R}$. Then

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

Hence, to complete the proof, we need to show that

[TABLE]

converges to $\alpha$ in probability and

[TABLE]

converges to $\beta$ in probability (note that $2^{-2k}=(b-a)\cdot N_{k}^{-1}$).

We have that $\left(W_{0}(t):t\in\mathbb{R}\right)$ and $\left(W_{0}(t+a)-W_{0}(a):t\in\mathbb{R}\right)$ have the same distribution. Then for $\widehat{W_{k}}(t)=2^{k}W_{0}(2^{-2k}t+a))-2^{k}W_{0}(a)$ we get that $\left(\widehat{W_{k}}(t):t\in\mathbb{R}\right)$ and $\left(W_{0}(t):t\in\mathbb{R}\right)$ have the same distribution by Lemma 13.8 for $r=2^{-k}$. Let $Y_{k}=2^{k}W_{0}(a)$. Then the distributions of $Y_{k}$ are absolutely continuous with respect to the Lebesgue measure and the density functions of $\left\{Y_{k}\right\}_{0}$ are uniformly bounded by Lemma 13.3. Thus the quantities in (13.15) and (13.16) are converging in probability to the desired limit by Proposition 13.7. ∎

Lemma 13.10**.**

Let $W_{0}(t)$ be a two-sided Brownian motion in $\mathbb{R}^{d}$ ($d\geq 3$) started at [math] and let $I\subseteq\mathbb{R}$ be a compact interval. Let

[TABLE]

Then $D$ is a random compact set and

[TABLE]

almost surely. In particular, $\lambda(D)=0$ almost surely.

Proof.

We have that

[TABLE]

hence almost surely the intersection of a compact set and a closed set and so $W_{0}(D)$ is almost surely a compact set. It follows from [19, Lemma 9.4] and the reversibility that $\dim_{H}W_{0}(D)\leq 1$ almost surely for $d=3$. It follows from [19, Theorem 9.1] and the Markov property that $D=\partial I$ almost surely for $d\geq 4$. So $\dim_{H}W_{0}(D)\leq 1$ almost surely. Since $W_{0}$ is almost surely a continuous function it follows that

[TABLE]

is a closed set and bounded because of the transience of the Brownian motion [19, Theorem 3.20], thus $D$ is compact. It follows from the fact that $\dim_{H}W_{0}(D)\leq 1$ almost surely and from Kaufman‘s dimension doubling theorem [19, Theorem 9.28] that

[TABLE]

almost surely. ∎

Lemma 13.11**.**

Let $W_{0}(t)$ be a two-sided Brownian motion in $\mathbb{R}^{d}$ ($d\geq 3$) started at [math], let $I\subseteq\mathbb{R}$ be a compact interval and let $\varepsilon>0$. Then almost surely there exist finitely many random open intervals $J_{1},\dots,J_{m}$ with dyadic endpoints such that

[TABLE]

and

[TABLE]

Proof.

Let $D$ be as in Lemma 13.10. Since $\lambda(D)=0$ and $D$ is compact almost surely we can cover $D$ with finitely many random open intervals $J_{1},\dots,J_{m}$ with dyadic endpoints such that $\sum_{i=1}^{m}\lambda(J_{i})<\varepsilon$. It is not hard to see that we can choose the open intervals $J_{1},\dots,J_{m}$ on a Borel measurable way, so we legitimately say random open intervals.

We have that $W_{0}(I)$ and $W_{0}(\mathbb{R}\setminus\left(\mathrm{int}I\bigcup\left(\cup_{i=1}^{m}J_{i}\right)\right)$ are disjoint almost surely due to the definition of $D$ (note that the endpoints of $I$ are contained in $D$). Thus

[TABLE]

almost surely because $W_{0}(I)$ is almost surely compact and $W_{0}(\mathbb{R}\setminus\left(\mathrm{int}I\bigcup\left(\cup_{i=1}^{m}J_{i}\right)\right)$ is almost surely closed. ∎

Remark 13.12*.*

It follows from [19, Corollary 3.19] that $W(\mathbb{R})\cap\{0\}=\emptyset$ almost surely. Hence for every deterministic interval $I\subseteq\mathbb{R}$ we have that

[TABLE]

almost surely. Similar holds for $W$ replaced by $W_{0}$ when $I\subseteq(0,\infty)$ because $W_{0}((0,\infty))\cap\{0\}=\emptyset$ almost surely.

Proposition 13.13**.**

Let $0\leq\alpha<\infty$ be as in Lemma 13.9. Let $W_{0}(t)$ be a two-sided Brownian motion in $\mathbb{R}^{d}$ ($d\geq 3$) started at [math]. Let $I=\cup_{n=1}^{N}I_{n}$ where $I_{1},\dots,I_{N}$ are disjoint compact intervals with positive dyadic endpoints. Then

[TABLE]

converges to $\alpha\lambda(I)$ in probability.

Proof.

To prove the statement of the proposition, by Lemma 2.9, it is enough to show that for every $\varepsilon>0$ and for every subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ we can find a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ such that

[TABLE]

It follows from Remark 13.12 that

[TABLE]

which we use throughout the proof instead of (13.17).

Let $\varepsilon>0$ be fixed and $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ be a subsequence of $\mathbb{N}$. Let $0\leq\beta<\infty$ be as in Lemma 13.9. For every compact interval $J$ with positive dyadic endpoints we have that $f_{k}^{J}$ converges to $\alpha\lambda(J)$ in probability and $g_{k}^{J}$ converges to $\beta\lambda(J)$ in probability by Lemma 13.9. Hence, by Lemma 2.8, we can find an event $H$ with $P(H)=1$ and a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ such that $f_{\beta_{k}}^{J}(\omega)$ converges to $\alpha\lambda(J)$ and $g_{\beta_{k}}^{J}(\omega)$ converges to $\beta\lambda(J)$ for every interval $J$ with positive dyadic endpoints for every outcome $\omega\in H$. We can further assume, by Lemma 13.11, that there exist $m_{j}(\omega)\in\mathbb{N}$ and finitely many open intervals $J_{1}^{j}(\omega),\dots,J_{m_{j}}^{j}(\omega)$ with dyadic endpoints for $j=1,\dots,N$ such that

[TABLE]

and

[TABLE]

for every $\omega\in H$.

Let $\omega\in H$ be fixed. If $k$ is large enough that $\mathrm{diam}(Q)<2^{-1}r_{j}$ for some $j$ and $Q\in\mathcal{Q}_{k}^{*}$ then whenever $Q\cap W_{0}^{\omega}(I_{j})\neq\emptyset$ then, by (13.19), either $Q\cap W_{0}^{\omega}(I_{j})\neq\emptyset$ for $I_{j}=[a_{j},b_{j}]$ and $Q\cap W_{0}^{\omega}(-\infty,a_{j})=\emptyset$ or $Q\cap W_{0}^{\omega}(J_{i}^{j}(\omega))\neq\emptyset$ for some $i$. Thus

[TABLE]

for large enough $k$, where $\overline{J_{i}}(\omega)$ is the closure of $J_{i}(\omega)$. Hence

[TABLE]

by (13.20). Since this holds for every $\omega\in H$ it follows that

[TABLE]

We have that

[TABLE]

and hence (13.18) follows from (13.21) and the fact that $\sum_{j=1}^{N}f_{\beta_{k}}^{I_{j}}(\omega)$ converges to $\alpha\lambda(I)$ on $H$. ∎

13.3 Occupation measure and limit measure

Throughout this subsection let $B_{0}(.)$ be a standard Brownian motion in $\mathbb{R}^{d}$ ($d\geq 3$) started at [math], let $B=\{B_{0}(t):t\in[0,\infty)\}$ be the range of $B_{0}$. Let $\tau$ be the occupation measure of $B_{0}(.)$, that is

[TABLE]

for every Borel set $A\subseteq\mathbb{R}^{d}$, i.e. the amount of time that the Brownian motion spends in $A$.

Lemma 13.14**.**

Let $Q\in\mathcal{Q}_{n}$ for some $n\in\mathbb{N}$. Then $\lambda(\partial B_{0}^{-1}(Q))=0$ almost surely.

Proof.

Since $B_{0}$ is almost surely continuous it follows that $\partial B_{0}^{-1}(Q)\subseteq B_{0}^{-1}(\partial Q)$ almost surely. Hence it is enough to prove that $\lambda\left(B_{0}^{-1}(\partial Q)\right)=0$. Let $B_{0}(t)=(B_{1}(t),\dots,B_{d}(t))$ for every $t\in[0,\infty)$. Then

[TABLE]

almost surely for every $i=1,\dots,d$ and $a\in\mathbb{R}$ by [19, Theorem 9.34]. For every side of $Q$ there exists $i\in\{1,\dots,d\}$ and $a\in\mathbb{R}$ such that the preimage of that side of $Q$ is contained in $B_{i}^{-1}(a)$. Thus it follows that

[TABLE]

because $Q$ has finitely many sides. ∎

Lemma 13.15**.**

Let $Q\in\mathcal{Q}_{n}$ for some $n\in\mathbb{N}$ such that $\mathrm{dist}(Q,0)>0$ and let $\varepsilon>0$ be fixed. Then almost surely there exist random sets $I^{-}=\cup_{n=1}^{N^{-}}I_{n}^{-}$ where $I_{1}^{-},\dots,I_{N^{-}}^{-}$ are random disjoint compact intervals with positive dyadic endpoints and $I^{+}=\cup_{n=1}^{N^{+}}I_{n}^{+}$ where $I_{1}^{+},\dots,I_{N^{+}}^{+}$ are random disjoint compact intervals with positive dyadic endpoints such that

[TABLE]

and $\lambda(I^{+}\setminus I^{-})<\varepsilon$.

Proof.

We have that $\lambda(\partial B_{0}^{-1}(Q))=0$ almost surely by Lemma 13.14. Hence almost surely there exist random sets $I^{-}=\cup_{n=1}^{N^{-}}I_{n}^{-}$ where $I_{1}^{-},\dots,I_{N^{-}}^{-}$ are random disjoint compact intervals with positive dyadic endpoints and $I^{+}=\cup_{n=1}^{N^{+}}I_{n}^{+}$ where $I_{1}^{+},\dots,I_{N^{+}}^{+}$ are random disjoint compact intervals with positive dyadic endpoints such that

[TABLE]

and $\lambda(I^{+}\setminus I^{-})<\varepsilon$. It is easy to see that we can choose the intervals on a measurable way. ∎

Lemma 13.16**.**

Let $Q\in\mathcal{Q}_{n}$ for some $n\in\mathbb{N}$ such that $\mathrm{dist}(Q,0)>0$ and let $G(x,y)$ be as in Remark 1.25. Then

[TABLE]

and

[TABLE]

Proof.

For every $Q=[a_{1},b_{1})\times\dots\times[a_{d}.b_{d})\in Q_{k}$ let $x_{Q}=(a_{1},\dots,a_{d})\in Q$. By applying Lemma 12.4 and Remark 12.7 to $A=[0,1)^{d}$ it follows that

[TABLE]

and

[TABLE]

It is easy to show that

[TABLE]

converges to $1$ uniformly on $Q$. Hence (13.23) follows from (13.25) and (13.24) follows from (13.26). ∎

Lemma 13.17**.**

Let $0\leq\alpha<\infty$ be as in Proposition 13.13, $G(x,y)$ be as in Remark 1.25 and $Q\in\mathcal{Q}_{n}$ for some $n\in\mathbb{N}$ such that $\mathrm{dist}(Q,0)>0$. Then

[TABLE]

converges to $\gamma\cdot\tau(Q)$ in probability as $k$ goes to $\infty$, where

[TABLE]

Proof.

Let

[TABLE]

then

[TABLE]

for every $S\in\mathcal{Q}_{k}$, $S\subseteq Q$ for every $k\geq n$. Thus

[TABLE]

[TABLE]

Similarly it can be shown that

[TABLE]

for

[TABLE]

To prove the statement of the lemma, by Lemma 2.9, it is enough to show that for every $\varepsilon>0$ and for every subsequence $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ of $\mathbb{N}$ we can find a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ such that

[TABLE]

Let $\varepsilon>0$ be fixed and $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ be a subsequence of $\mathbb{N}$. For every $I=\cup_{n=1}^{N}I_{n}$ where $I_{1},\dots,I_{n}$ are disjoint compact intervals with positive dyadic endpoints we have that

[TABLE]

converges to $\alpha\lambda(I)$ in probability by Proposition 13.13. Hence, by Lemma 2.8, we can find an event $H$ with $P(H)=1$ and a subsequence $\left\{\beta_{k}\right\}_{k=1}^{\infty}$ of $\left\{\alpha_{k}\right\}_{k=1}^{\infty}$ such that $h_{\beta_{k}}^{I}(\omega)$ converges to $\alpha\lambda(I)$ for every $I=\cup_{n=1}^{N}I_{n}$ where $I_{1},\dots,I_{n}$ are disjoint compact intervals with positive dyadic endpoints for every outcome $\omega\in H$. We can further assume, by Lemma 13.15, that for every $\omega\in H$ there exist $N^{-}(\omega),N^{+}(\omega)\in\mathbb{N}$ and sets $I^{-}(\omega)=\cup_{n=1}^{N^{-}(\omega)}I_{n}^{-}(\omega)$ where $I_{1}^{-}(\omega),\dots,I_{N^{-}}^{-}(\omega)$ are disjoint compact intervals with positive dyadic endpoints and $I^{+}(\omega)=\cup_{n=1}^{N^{+}(\omega)}I_{n}^{+}(\omega)$ where $I_{1}^{+}(\omega),\dots,I_{N^{+}}^{+}(\omega)$ are disjoint compact intervals with positive dyadic endpoints such that

[TABLE]

and

[TABLE]

Let $\omega\in H$ be fixed. Then

[TABLE]

by (13.28) and (13.31). Thus

[TABLE]

[TABLE]

by Lemma 13.16, the fact that $h_{\beta_{k}}^{I^{+}}(\omega)$ converges to $\lambda(I^{+})$, (13.27) and that $\lambda(I^{+})\leq\lambda(B_{0}^{-1}(Q))+\varepsilon/2\gamma=\tau(Q)+\varepsilon/2\gamma$ by (13.32) and the definition of $\tau$, (13.22). Similarly we can show that

[TABLE]

Hence

[TABLE]

because (13.33) and (13.34) hold for every $\omega\in H$ and $P(H)=1$. Thus (13.30) follows. ∎

Theorem 13.18**.**

Let $\nu(A)=\intop_{A}\left\|x\right\|^{2-d}\mathrm{d}x$ for every Borel set $A\subseteq\mathbb{R}^{d}$. Then $\nu$ is a locally finite Borel measure, $\nu=\nu_{R}$ and $\mathcal{C}(\nu)=\frac{1}{c(d)}\tau$ almost surely where $c(d)$ is as in (1.26).

Proof.

Clearly $\nu$ is a measure. For every $y\in\mathbb{R}^{d}$ such that $\left\|y\right\|>r>0$ for some $r$ we have that $\nu(B(y,r))\leq\left(\left\|y\right\|-r\right)^{2-d}\lambda(B(y,r))$. By the argument in the last paragraph of [17, page 109] it follows that

[TABLE]

and

[TABLE]

Hence $\nu$ is a locally finite measure and $\nu_{R}=\nu$ by Proposition 1.9.

The conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi(x,y)=\left\|x-y\right\|^{2-d}$ by Theorem 1.22. Let $\gamma$ be as in Lemma 13.17. It follows from Lemma 13.17 that $\mathcal{C}_{k}(\nu)(Q)$ converges to $\gamma\cdot\tau(Q)$ in probability for every $n\in\mathbb{N}$ for every $Q\in\mathcal{Q}_{n}$ such that $\mathrm{dist}(Q,0)>0$. By Property ii.) and iv.) of Definition 1.11 we have that $\mathcal{C}(\nu)(Q)=\gamma\cdot\tau(Q)$ almost surely for every $n\in\mathbb{N}$ for every $Q\in\mathcal{Q}_{n}$ such that $\mathrm{dist}(Q,0)>0$. Hence it follows that

[TABLE]

almost surely for every $n\in\mathbb{N}$ for every $Q\in\mathcal{Q}_{n}$ because $Q$ can be written as a countable union of boxes $Q_{i}\in\mathcal{Q}_{n_{i}}$ ($i\in\mathbb{N}$) such that $\mathrm{dist}(Q_{i},0)>0$. It follows from Property v.) of Definition 1.11 that $\mathcal{C}(\nu)(\{0\})=0$ almost surely. The set $\{\{0\}\}\cup\bigcup_{n=1}^{\infty}\mathcal{Q}_{n}$ forms a semiring that generates the Borel $\sigma$-algebra hence it follows by Proposition 2.33 that

[TABLE]

almost surely.

By Property ii.) of Definition 1.11 it follows that

[TABLE]

On the other hand

[TABLE]

by [19, Theorem 3.32] and [19, Theorem 3.33]. Thus $\gamma=c(d)^{-1}$ because the integral is positive and finite, see (13.35). ∎

Theorem 13.19**.**

We have that $\mathrm{d}\mathcal{C}(\lambda)=\frac{1}{c(d)}\left\|x\right\|^{d-2}\mathrm{d}\tau$ almost surely where $c(d)$ is as in (1.26).

Proof.

The conditional measure $\mathcal{C}(\lambda)$ of $\lambda$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi(x,y)=\left\|x-y\right\|^{2-d}$ by Theorem 1.22. The statement follows from Theorem 13.18 and Property x.)* of Definition 1.11. ∎

Remark 13.20*.*

It turns out in the proof of Theorem 13.18 that $\gamma=c(d)^{-1}$ and $\gamma=\alpha\cdot C_{G}([0,1]^{d})$ by (13.27) where $G(x,y)=c(d)\left\|x-y\right\|^{2-d}$. Hence it follows that

[TABLE]

for $\varphi(x,y)=\left\|x-y\right\|^{2-d}$.

14 Conditional measure on the percolation limit sets of trees

In this section we move to other metric spaces than $\mathbb{R}^{d}$ to study the conditional measure of measures on the boundary of a tree when $B$ is a percolation limit set.

Definition 14.1**.**

Let $T=(V,H,\zeta)$ be a countable graph with vertex set $V$, edge set $H$ and a special vertex $\{\zeta\}$, that we call the* root of* $T$. We say that $T$* is a rooted tree with root $\zeta$* if the following hold:

i.) for every $v,w\in V$ there exists a unique self-avoiding finite path from vertex $v$ to vertex $w$,

ii.) the degree of every vertex $v\in V$ is finite,

iii.) let $T_{0}=\{\zeta\}$ and for every positive integer $n$ let $T_{n}$ be the collection of vertices $v\in V$ such that there exists a unique self-avoiding path of length $n$ from $\zeta$ to $v$. Then $\left\{(v,w)\in V:w\in T_{n+1}\right\}\neq\emptyset$ for every $v\in T_{n}$ ($n=0,1,\dots$).

We call an infinite self-avoiding path starting at $\zeta$ a ray. We denote the set of rays by $\partial T$. For every vertex $v\in V$ let $\left|v\right|$ be the unique $n\in\mathbb{N}$ such that $v\in T_{n}$. For two rays $x,y\in\partial T$ let $x\wedge y\in V$ be the unique vertex such that both $x$ and $y$ visit $x\wedge y$ and for every $v\in T_{n}$ for every $n>\left|x\wedge y\right|$ at most one of $x$ and $y$ visits $v$. We define a metric on $\partial T$ by

[TABLE]

Then $X=\partial T$ is a compact separable metric space that is homeomorphic to the Cantor set. For two vertices $v,w\in V$ let $v\wedge w\in V$ be the unique vertex such that the unique self-avoiding paths from $\zeta$ to $v$ and $\zeta$ to $w$ both visit $v\wedge w$ and for every $z\in T_{n}$ for every $n>\left|x\wedge y\right|$ at most one of the unique self-avoiding paths from $\zeta$ to $v$ and $\zeta$ to $w$ visits $v$.

For the rest of this section let $\alpha>0$ be fixed. Let $\varphi_{\alpha}(r)=r^{-\alpha}$ for $r>0$ and so

[TABLE]

for $x,y\in\partial T$. Then for every $\delta>0$ we have that (1.16) holds for $c_{2}=(1+2\delta)^{\alpha}$ and $c_{3}=0$. Also (1.17) holds.

For a vertex $v\in V$ we denote by $[v]$ the set of rays of $\partial T$ that goes through vertex $v$. Let

[TABLE]

Then $\mathcal{Q}_{k}$ is a sequence of finite families of Borel subsets of $\partial T$ such that $Q\cap S=\emptyset$ for $Q,S\in\mathcal{Q}_{k}$, for all $k\in\mathbb{N}$. It is easy to see that (1.11) holds. For every $v\in T_{k}$ we have that $\mathrm{diam}([v])=2^{-k}$, hence (1.10) and (1.19) hold. For $v,w\in T_{k}$, $v\neq w$ we have that $\mathrm{dist}([v],[w])\geq 2^{-(k-1)}$, and so

[TABLE]

for every $Q\in\mathcal{Q}_{k}$ for $1/2<\delta<1$. Thus (1.18) holds for such $\delta$ and $M_{\delta}=1$.

Let

[TABLE]

be a probability parameter. For every vertex $v\in V$ let $Y(v)$ be a Bernoulli variable with parameter $p$, such that $Y(v)$ ($v\in T$) are mutually independent. Let

[TABLE]

be a random compact set. Let

[TABLE]

be a random compact set which we call the percolation limit set.

The following result is due to Lyons, see for example [19, Theorem 9.17]

Proposition 14.2**.**

For every compact set $K\subseteq\partial T$

[TABLE]

In particular, $P(B\cap K\neq\emptyset)=0$ if $C_{\alpha}(K)=0$.

By Proposition 14.2 and (14.1) it follows that (1.13) and (1.22) holds for $a=1$.

For $v,w\in V$ such that $[v]$ and $[w]$ are disjoint, we have that

[TABLE]

for $k=\left|v\wedge w\right|$, hence if further $C_{\alpha}([v])>0$ and $C_{\alpha}([w])>0$ then

[TABLE]

[TABLE]

[TABLE]

Thus (1.23) holds for $c=1$ for every disjoint $Q$ and $S$ and so for every $\delta>0$. We have discussed above that the assumptions of Section 1.1.2 are satisfied.

It follows from (14.2) that

[TABLE]

for every $x,y\in\partial T_{0}=\bigcap_{k=1}^{\infty}(\bigcup_{Q\in\mathcal{Q}_{k}}Q)$, $x\neq y$ and $F(x,y)=0$ otherwise, see (1.19). As we established above the conditions of Theorem 1.21 and Theorem 1.28 are satisfied and hence we can conclude Theorem 14.4.

Remark 14.3*.*

If $I_{\alpha}(\nu)<\infty$ and $\nu(Q)>0$ then $C_{\alpha}(Q)>0$ because $I_{\alpha}(\nu|_{Q})<\infty$ and so $\mathrm{supp}\nu\subseteq\partial T_{0}$. Hence if $\nu(\partial T_{0})=0$ then $\nu_{R}=\nu_{\varphi_{\alpha}R}=0$. On the other hand if $\nu$ is a finite Borel measure such that $\nu(\partial T_{0})=0$ then $\mathcal{C}_{k}(\nu)$ converges to [math] in $\mathcal{L}^{1}$ by Lemma 5.4. Hence for such $\nu$ the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi_{\alpha}$ and $\mathcal{C}(\nu)=0$ almost surely. Thus the assumption that $\nu(\partial T\setminus\partial T_{0})=0$ is eliminated in the following Theorem.

Theorem 14.4**.**

Let $\nu$ be a finite, Borel measure on $\partial T$. Then the conditional measure $\mathcal{C}(\nu)$ of $\nu$ on $B$ exists with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi_{\alpha}$ and if $\tau$ is a finite, Borel measure on $\partial T$ then

[TABLE]

for every $f:X\times X\longrightarrow\mathbb{R}$ Borel function with $\intop\intop\varphi_{\alpha}(x,y)\left|f(x,y)\right|\mathrm{d}\nu_{R}(x)\mathrm{d}\tau_{R}(y)<\infty$. Moreover,

[TABLE]

Remark 14.5*.*

By Theorem 8.1 if $A\subseteq X$ is a Borel set such that $I_{\alpha}(\nu|_{A})<\infty$ then $\mathcal{C}_{k}(\nu)(A)$ converges to $\mathcal{C}(\nu)(A)$ in $\mathcal{L}^{2}$ (recall that $\nu|_{A}(T\setminus\partial T_{0})=0$ by Remark 14.3).

14.1 Random multiplicative cascade measure as conditional measure

Throughout this section for a finite Borel measure $\nu$ on $\partial T$ let $\mathcal{C}(\nu)$ be the conditional measure of $\nu$ on $B$ with respect to $\mathcal{Q}_{k}$ ($k\geq 1$) with regularity kernel $\varphi_{\alpha}$ (which exists by Theorem 14.4). Let

[TABLE]

let $\mathcal{F}_{k}$ be the $\sigma$-algebra generated by the events $\left\{Q\cap B_{k}\neq\emptyset\right\}_{Q\in\mathcal{S}_{k}}$ and for a finite Borel measure $\nu$ on $\partial T$ let

[TABLE]

Note that $\mu_{n}^{\nu}$ is the conditional measure of $\nu$ on $B_{n}$ with respect to $\mathcal{S}_{k}$ ($k\geq 1$) with regularity kernel $\varphi(x,y)=1$.

Proposition 14.6**.**

Let $\nu$ be a finite Borel measure on $\partial T$. There exists a random, finite Borel measure $\mu^{\nu}$ on $\partial T$ with the following properties:

1.) $\mu_{k}^{\nu}$ weakly converges to $\mu^{\nu}$ almost surely,

2.) for a countable collection of Borel sets $A_{n}\subseteq\partial T$ ($n\in\mathbb{N}$) we have that $\mu_{k}^{\nu}(A_{n})$ converges to $\mu^{\nu}(A_{n})$ for every $n\in\mathbb{N}$ as $k$ goes to $\infty$ almost surely,

3.) $\mu^{\nu}=\mu^{\nu_{R}}$ almost surely for $\nu_{R}=\nu_{\varphi_{\alpha}R}$,

4.) $\mu^{\nu_{\perp}}=0$ almost surely for $\nu_{\perp}=\nu_{\varphi_{\alpha}\perp}$,

5.) $\mu^{\nu}=\sum_{i=1}^{\infty}\mu^{\nu_{i}}$ if $\nu=\sum\nu_{i}$,

6.) $E\left(\mu^{\nu}(A)\right)\leq\nu(A)$ for every Borel set $A\subseteq\partial T$.

Proof.

It is easy to check that the sequence of random measures $\mu_{k}^{\nu}$ is a $T$-martingale with respect to the filtration $\mathcal{F}_{k}$. Hence Property 1.) and 2.) follows from [12, Theorem 1]. Property 6.) follows from Property 2.) and the nonnegative martingale convergence theorem [8, Theorem 5.2.9]. Property 5.) follows from Proposition 3.14.

If $C_{\alpha}(K)=0$ for a compact set $K\subseteq\partial T$ then $P(B\cap K\neq\emptyset)=0$ by Proposition 14.2. Hence, via an argument that is similar to the proof of Theorem 5.2, it can be shown that $\mu_{k}^{\nu_{\perp}}(\partial T)$ converges to [math] in probability. It implies that Property 3.) and 4.) hold. ∎

Remark 14.7*.*

When $T$ is an $m$-ary tree for some $m\in\mathbb{N}$ and $\nu$ is the uniform measure on $\partial T$ then $\mu^{\nu}$ is the random multiplicative cascade measure with weight variables $Y(v)/p$ for $v\in T$.

Lemma 14.8**.**

Let $\nu$ be a finite Borel measure on $\partial T$ such that $I_{\alpha}(\nu)<\infty$. Then

[TABLE]

Proof.

We have that if $\nu(Q)>0$ then $\nu(Q)^{2}/I_{\alpha}(\nu|_{Q})\leq C_{\alpha}(Q)$. It follows from Proposition 14.2 that $C_{\alpha}(Q)\cdot p^{-k}\leq P\left(Q\cap B\neq\emptyset\mid Q\cap B_{k}\neq\emptyset\right)$ for every $Q\in\mathcal{Q}_{k}$. Hence

[TABLE]

[TABLE]

Since $I_{\alpha}(\nu)<\infty$ the statement follows because

[TABLE]

by Fubini‘s theorem. ∎

Lemma 14.9**.**

Let

[TABLE]

for $Q,S\in\mathcal{Q}_{k}$. Then

[TABLE]

Proof.

We have that $P(Q\cap B_{k}\neq\emptyset)=P(S\cap B_{k}\neq\emptyset)=p^{k}$. Since $Q\in\mathcal{Q}_{k}$ it follows that

[TABLE]

and similarly for $S$. Then

[TABLE]

[TABLE]

Given $\mathcal{F}_{k}$ if $Q\neq S$ then we have that $I_{Q\cap B\neq\emptyset}$ and $I_{S\cap B\neq\emptyset}$ are independent, hence $E\left(f(Q,S)\mid\mathcal{F}_{k}\right)=0$.

Let $q=P(Q\cap B\neq\emptyset\mid Q\cap B_{k}\neq\emptyset)$. If $Q=S$ then

[TABLE]

[TABLE]

Hence

[TABLE]

∎

Proposition 14.10**.**

Let $\nu$ be a finite Borel measure on $\partial T$ such that $I_{\alpha}(\nu)<\infty$. Then

[TABLE]

for every Borel set $A\subseteq\partial T$.

Proof.

If $\nu(A)=0$ then the proof is trivial so we assume that $\nu(A)>0$. Without the loss of generality we can assume that $A=\partial T$ otherwise we replace $\nu$ by $\nu|_{A}$. By Remark 14.3 we have that $\nu(Q)=0$ for every $Q\in\mathcal{S}_{k}\setminus\mathcal{Q}_{k}$. Hence by Lemma 14.9

[TABLE]

[TABLE]

Thus the statement follows from Lemma 14.8. ∎

Proposition 14.11**.**

Let $\nu$ be a finite Borel measure on $\partial T$ such that $I_{\alpha}(\nu)<\infty$. Then $\mu^{\nu}=\mathcal{C}(\nu)$ almost surely.

Proof.

We have that $\mathcal{C}_{k}(\nu)(Q)$ converges to $\mathcal{C}(\nu)(Q)$ in $\mathcal{L}^{2}$ for every $Q\in\mathcal{S}_{k}$ by Remark 14.5. Hence it follows by Proposition 14.10 that $\mu_{k}^{\nu}(Q)$ converges to $\mathcal{C}(\nu)(Q)$ in $\mathcal{L}^{2}$. So by Property 2.) of Proposition 14.6 it follows that $\mu^{\nu}(Q)=\mathcal{C}(\nu)(Q)$ for every $Q\in\mathcal{S}_{k}$ almost surely. Since $\cup_{k=1}^{\infty}\mathcal{S}_{k}$ is a semiring that generates the Borel $\sigma$-algebra of $\partial T$ it follows by Proposition 2.33 that $\mu^{\nu}=\mathcal{C}(\nu)$ almost surely. ∎

As we noted in Remark 14.7 the random measure $\mu^{\nu}$ is a multiplicative cascade measure when $\nu$ is the uniform measure. The next theorem states that $\mu^{\nu}$ is the conditional measure $C(\nu)$.

Theorem 14.12**.**

Let $\nu$ be a finite Borel measure on $\partial T$. Then $\mu^{\nu}=\mathcal{C}(\nu)$ almost surely.

Proof.

By Proposition 1.7 there exists a sequence of finite Borel measures $\nu_{i}$ such that $\nu=\nu_{\varphi_{\alpha}\perp}+\sum_{i=1}^{\infty}\nu_{i}$ and $I_{\alpha}(\nu_{i})<\infty$ for every $i\in\mathbb{N}$. It follows from Proposition 14.11 that $\mu^{\nu_{i}}=\mathcal{C}(\nu_{i})$ for every $i\in\mathbb{N}$ almost surely. It follows from Property 4.) of Proposition 14.6 that $\mu^{\nu_{\varphi_{\alpha}\perp}}=0$ almost surely and it follows from Property vii.) of Definition 1.11 that $\mathcal{C}(\nu_{\varphi_{\alpha}\perp})=0$ almost surely. Thus it follows from Property 5.) of Proposition 14.6 and Property ix.) of Definition 1.11 that

[TABLE]

almost surely. ∎

$\mathbf{Acknowledgements}$

The author was partially supported by the ERC grant (grant number 306494) and by the ERC grant (grant number 772466). The author would like to thank Xiong Jin, Zemer Kosloff, Georg Berschneider, Yuval Peres, David Simmons, Mike Hochman, Benjamin Weiss, Kenneth Falconer, Bálint Tóth and Gábor Pete for the many useful discussion.