Measures and integrals in conditional set theory

TL;DR

This paper develops foundational results in conditional measure theory, demonstrating how kernels and distributions can be represented by measures within a conditional set framework, extending classical representation theorems.

Contribution

It introduces basic results in conditional measure theory and extends representation theorems to non-separable spaces within this framework.

Findings

Established foundational results in conditional measure theory.

Proved that kernels and distributions are representable by measures in conditional set theory.

Extended classical representation results to broader spaces.

Abstract

The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In particular, this extends the usual representation results for separable spaces.

Equations

$$\mathbb{E}[f(\xi )|\mathcal{G}](\omega )=\int f(x)\kappa (\omega ,dx)\text{almost surely,}$$

$$\text{st}(V):=\left\{\sum x_k|A_k:(A_k)\text{ partition, }(x_k)\text{ sequence in }V\right\}.$$

$$\sum V_k|A_k:=\left\{\sum x_k|A_k:x_k\in V_k\text{ for each }k\right\}$$

$$\prod X_i:=\left\{x:\Omega \to \prod E_i:x(\omega )=(x_i(\omega )),x_i\in X_i\text{ for each }i\right\}$$

$$\sum (V_k|B_k)|A_k:=(\sum V_k|A_k)|{\cup }_k(A_k\cap B_k)\in \mathcal{P}(X),$$

$$V|A⊑W|B\text{ if and only if }A\subseteq B\text{ and }V|A\subseteq W|A.$$

$$V:=\left\{\sum x_k|B_k+x_0|A^c:\text{ for all }k\text{ there is }i\text{ s.t. }{B}_k\subseteq A_i\text{ and }{x}_k\in V_i\text{ and }(B_k)\text{ is a partition of }A\right\}.$$

$${\bigsqcup }_i{V}_i|A_i:=V|A$$

$$\mathcal{G}=\{A\in \mathcal{F}:A\subseteq \underset{i}{\inf }{A}_i\text{ and }{\cap }_i{V}_i|A\ne \varnothing \}.$$

$${\sqcap }_i{V}_i|A_i:=W|A_{\ast }$$

$$(V|A{)}^{⊏}:=\bigsqcup \{W|B\in \mathcal{P}(X):W|B\sqcap V|A=\{\ast \}\}.$$

$$(V|A{)}^{⊏}=Z|B_{\ast }+X|A^c,$$

$$B_{\ast }=\sup \{A^{\prime }\in \mathcal{F}:A^{\prime }\subseteq A,V|A^{\prime }\ne X|A^{\prime }\}$$

$$Z=\{\begin{array}{l}\{x\in X:x|A^{\prime }\notin V|A^{\prime }\text{ for all }{A}^{\prime }\in {\mathcal{F}}_{+}\text{ with }{A}^{\prime }\subseteq B_{\ast }\},\text{if }{B}_{\ast }\in {\mathcal{F}}_{+},\\ X,\text{ else}.\end{array}$$

$$(\sum (V_k|A_k)|D_k{)}^{⊏}=((\sum V_k|D_k)|A{)}^{⊏}=U|B_{\ast }+X|A^c,$$

$$(\sum (U_k|B_{k,\ast }+X|A_k^c)|D_k)|{\cup }_k(A_k^c\cap D_k)=X|A^c.$$

$$U|B_{\ast }=(\sum U_k|D_k)|{\cup }_k(B_{k,\ast }\cap D_k).$$

$$C_{\ast }:=\sup \{A^{\prime }\in \mathcal{F}:A^{\prime }\subseteq A,\text{ there is }x\in X\text{ such that }f(x)|A^{\prime }\in W|A^{\prime }\}.$$

$$V:=\{x\in X:f(x)|C_{\ast }\in W|C_{\ast }\}.$$

$$f^{-1}(W|A):=V|C_{\ast }.$$

$$x\mapsto (\sum f_k|A_k)(x):=\sum f_k(x)|A_k,$$

$$\mathcal{Z}=\{V|A\in \mathcal{P}(Y):f^{-1}(V|A)\in \mathcal{X}\}$$

$$\mu (\sum (V_k|B_k)|A_k)=\sum \mu (V_k|B_k)|A_k$$

$${\delta }_x(V|A):=1|B+0|B^c,$$

$$\mu (V|A\bigsqcup W|B)$$

$$\mu (W|B)$$

$${\mu }^{\ast }(W|B\sqcap V|A)+{\mu }^{\ast }(W|B\sqcap (V|A{)}^{⊏})={\mu }^{\ast }(W|B)\text{ for all }W|B\in \mathcal{P}(X).$$

$${\mu }^{\ast }(Z|C)={\mu }^{\ast }(Z|C\sqcap W|B)+{\mu }^{\ast }(Z|C\sqcap (W|B{)}^{⊏})\text{ for all }Z|C\in \mathcal{P}(X).$$

$${\mu }^{\ast }(Z|C)$$

$$+{\mu }^{\ast }(Z|C\sqcap (V|A{)}^{⊏}\sqcap W|B)+{\mu }^{\ast }(Z|C\sqcap (V|A{)}^{⊏}\sqcap (W|B{)}^{⊏}).$$

$${\mu }^{\ast }(Z|C\sqcap (V|A\bigsqcup W|B))={\mu }^{\ast }(Z|C\sqcap V|A\sqcap W|B)+{\mu }^{\ast }(Z|C\sqcap V|A\sqcap W|B^{⊏})+{\mu }^{\ast }(Z|C\sqcap V|A^{⊏}\sqcap W|B).$$

Full text

A.J. and M.K. gratefully acknowledge financial support from DFG project KU 2740/2-1. The authors would like to thank an anonymous referee for a careful reading of the manuscript and helpful comments and suggestions.

1. Introduction

A random variable $\xi$ is a measurable function from a probability space $(\Omega,\mathscr{F},\mathbb{P})$ into a measurable space $(E,\mathcal{E})$. Given a sub-$\sigma$-algebra $\mathscr{G}\subseteq\mathscr{F}$, the conditional distribution of $\xi$ given $\mathscr{G}$, i.e. the quantity $\mathbb{E}[1_{\{\xi\in A\}}|\mathscr{G}](\omega)$ for $A\in\mathcal{E}$ and $\omega\in\Omega$, where $\mathbb{E}[\cdot|\mathscr{G}]$ denotes conditional expectation, can be computed via a probability kernel $\kappa\colon\Omega\times\mathcal{E}\to[0,1]$ whenever $(E,\mathcal{E})$ is a standard Borel space. In particular, one has

[TABLE]

for every measurable function $f\colon E\to\mathbb{R}$ such that $f(\xi)$ is integrable. By a conditional measure theory, one can extend the previous representation results to random variables with values in an arbitrary measurable space. More precisely, we will show that a conditional distribution corresponds to a real-valued measure in conditional set theory and that $\mathbb{E}[f(\xi)|\mathscr{G}]$ can be computed with the help of a conditional Lebesgue integral. Conditional measure theory suggests a natural formalism to study kernels and conditional distributions in a purely measure-theoretic context without unnecessary topological assumptions such as separability.

We will differ from the abstract setting in [9], where conditional set theory is developed relative to an arbitrary complete Boolean algebra. Motivated by the aforementioned applications, we construct a conditional measure theory relative to the complete Boolean algebra obtained from a probability space by quotienting out null sets. In accordance with this aim, we restrict attention to a class of conditional sets which are associated to spaces of random variables stable under countable concatenations. By a straightforward generalization, one can extend all results in this article to the abstract setting in [9].

Our contribution

We establish basic results in conditional measure theory, e.g. a $\pi$-$\lambda$-theorem, a Carathéodory extension theorem, construction of a Lebesgue measure and a Lebesgue integral, see Section 3. These results are applied to connect kernels and conditional distributions with measures in a conditional theory in Section 4. A conditional version of some standard theorems in measure theory is proved in Section 5.

Related literature

A conditional set is an abstract set-like structure, see [9] for a thorough introduction. Conditional set theory is closely related to Boolean-valued models and topos theory, see [18]. Basic results for Riemann integration in a Boolean-valued model are studied in [24, Chapter 2]. See [6, 11, 12, 15, 20, 23] for further results in conditional analysis and conditional set theory. For applications of conditional analysis, we refer to [2, 3, 5, 7, 8, 10, 13, 14, 17, 19]. See [4, 16] for other related results.

2. Preliminaries

Throughout, we fix a complete probability space $(\Omega,\mathscr{F},\mathbb{P})$, and identify $A,B\in\mathscr{F}$ whenever $\mathbb{P}(A\Delta B)=0$, where $\Delta$ denotes symmetric difference. In particular, $A\subseteq B$ is understood in the almost sure (a.s.) sense. Let $\mathscr{F}_{+}$ denote the set of all $A\in\mathscr{F}$ with $\mathbb{P}(A)>0$. For a function $x$ on $\Omega$ with values in an arbitrary set $E$ and $A\in\mathscr{F}$, we denote by $x|A\colon A\to E$ the restriction of $x$ to $A$. If $x,y\colon\Omega\to E$ are functions and $A,B\in\mathscr{F}$, then we always identify $x|A$ with $y|B$ whenever $\mathbb{P}(A\Delta B)=0$ and $x(\omega)=y(\omega)$ for a.a. $\omega\in A\cap B$. A partition is a countable family $(A_{k})$ of measurable subsets of $\Omega$ which is pairwise disjoint and such that $\cup_{k}A_{k}=\Omega$. Given a sequence of functions $x_{k}\colon\Omega\to E$ and a partition $(A_{k})$, we denote by $\sum x_{k}|A_{k}$ the unique function $x\colon\Omega\to E$ with the property that $x|A_{k}=x_{k}|A_{k}$ for all $k$. For a set $X$ of functions $x\colon\Omega\to E$ and $A\in\mathscr{F}$, let $X|A:=\{x|A\colon x\in X\}$, and define $(X|A)|B:=X|A\cap B$ for all $B\in\mathscr{F}$. Notice that $X|\emptyset=\{\ast\}$ is a singleton and trivially $X|\Omega=X$.

Denote by $\bar{L}^{0}=\bar{L}^{0}(\Omega,\mathscr{F},\mathbb{P})$ the set of all measurable functions $x\colon\Omega\to[-\infty,\infty]$, and by $L^{0}$ its subset of all real-valued functions. On $\bar{L}^{0}$ consider the complete order $x\leq y$ a.s. Recall that on $L^{0}$ this order is Dedekind complete. Let $\bar{L}^{0}_{+}:=\{x\in\bar{L}^{0}\colon x\geq 0\}$, $L^{0}_{+}:=\{x\in L^{0}\colon x\geq 0\}$ and $L^{0}_{++}:=\{x\in L^{0}\colon x>0\}$. For an arbitrary subset $\mathscr{G}\subseteq\mathscr{F}$, its supremum w.r.t. almost sure inclusion is the measurable set $A\in\mathscr{F}$ (unique mod a.s. identification) such that $1_{A}=\sup\{1_{A^{\prime}}\colon A^{\prime}\in\mathscr{G}\}$. Similarly, we define the infimum of $\mathscr{G}$, and denote them by $\sup\{A^{\prime}\colon A^{\prime}\in\mathscr{G}\}$ and $\inf\{A^{\prime}\colon A^{\prime}\in\mathscr{G}\}$, respectively. Throughout, all order relations between extended real-valued random variables are understood in the a.s. sense.

Definition 2.1**.**

A set $X$ of functions on $\Omega$ with values in a fixed image space $E$ is said to be stable under countable concatenations, or stable for short, if $X\neq\emptyset$ and $\sum x_{k}|{A_{k}}\in X$ for every partition $(A_{k})$ and each sequence $(x_{k})$ in $X$.

The spaces $\bar{L}^{0}$, $L^{0}$, $\bar{L}^{0}_{+}$, $L^{0}_{+}$ and $L^{0}_{++}$ are stable sets of functions on $\Omega$. We provide more examples which are important in the sequel.

Examples 2.2**.**

Let $E$ be a nonempty set, $(x_{k})$ a sequence in $E$ and $(A_{k})$ a partition. Let $\sum x_{k}|A_{k}$ denote the function on $\Omega$ with value $x_{k}$ on $A_{k}$ for each $k$ and call it a step function with values in $E$. The set of all step functions with values in $E$ is a stable set of functions on $\Omega$ and is denoted by $L^{0}_{s}(E)$. In particular, $L^{0}_{s}(\mathbb{N})$ denotes the set of all step functions with values in the natural numbers.

• 2)

Let $(E,\mathscr{E})$ be an arbitrary measurable space. The set of all measurable functions on $\Omega$ with values in $E$ is stable and is denoted by $L^{0}_{m}(E)$.

• 3)

If $E$ is a topological space and $\mathscr{E}$ its Borel $\sigma$-algebra. The set of all strongly measurable111Strongly measurable means Borel measurable and essentially separably-valued. functions on $\Omega$ with values in $E$ is stable and is denoted by $L^{0}(E)$.

• 4)

Every nonempty order-bounded subset of $L^{p}$ for $0\leq p\leq\infty$ is stable.

• 5)

Given a nonempty subset $V$ of a stable set of functions on $\Omega$, let

[TABLE]

We call $\text{st}(V)$ the stable hull of $V$.

• 6)

Let $X$ be a stable set of functions on $\Omega$, $(V_{k})$ a sequence of stable subsets of $X$ and $(A_{k})$ a partition. Then

[TABLE]

is a stable subset of $X$.

• 7)

Let $X_{i}$ be a stable set of functions on $\Omega$ with image spaces $E_{i}$ for each $i$ in an index. Their Cartesian product

[TABLE]

is a stable set of functions on $\Omega$.

Remark 2.3**.**

In general, $L^{0}(E)$ is strictly smaller than $L^{0}_{m}(E)$ for a topological space $E$. However, if $(E,\mathscr{E})$ is a standard Borel space, then $L^{0}(E)$ and $L^{0}_{m}(E)$ coincide.

We have a close connection between stable sets of functions on $(\Omega,\mathscr{F},\mathbb{P})$ and conditional sets of the associated measure algebra which will allows us to apply the machinery of conditional set theory [9]. We start by defining an analog of the conditional power set.

Definition 2.4**.**

Let $X$ be a stable set of functions on $\Omega$. Let $\mathcal{P}(X)$ denote the collection of all conditional subsets $V|A$, where $V$ is a stable subset of $X$ and $A\in\mathscr{F}$. A countable concatenation in $\mathcal{P}(X)$ is defined by

[TABLE]

for a sequence $(V_{k}|B_{k})$ and a partition $(A_{k})$, where $\sum V_{k}|A_{k}$ is defined in (2.1). A subset of $\mathcal{P}(X)$ is called a stable collection if it is nonempty and closed under countable concatenations.

In the present context, the conditional inclusion relation [9, Definition 2.9] reads as follows.

Definition 2.5**.**

The conditional inclusion relation on $\mathcal{P}(X)$ is the binary relation

[TABLE]

An important result in conditional set theory which will be helpful for the construction of a conditional measure theory is the following theorem which is proved in [9, Theorem 2.9].

Theorem 2.6**.**

The ordered set $(\mathcal{P}(X),\sqsubseteq)$ is a complete complemented distributive lattice.

Proof.

For the sake of completeness, we provide the main constructions of the proof in the present setting. Notice that $X$ is the greatest and $X|\emptyset=\{\ast\}$ the least element in $\mathcal{P}(X)$. Let $(V_{i}|A_{i})$ be a nonempty family in $\mathcal{P}(X)$. We construct its supremum and infimum w.r.t. $\sqsubseteq$. Put $A:=\sup_{i}A_{i}$ and fix some $x_{0}\in X$. Define

[TABLE]

The conditional subset

[TABLE]

is the supremum of $(V_{i}|A_{i})$. As for its infimum, let

[TABLE]

Put $A_{\ast}=\sup\mathscr{G}$. We show that $A_{\ast}$ is attained. Let $(A_{k})$ be a sequence in $\mathscr{G}$ such that $A_{\ast}=\sup_{k}A_{k}$. Define $B_{1}=A_{1}$ and $B_{k}=A_{k}\cap(B_{1}\cup\ldots\cup B_{k-1})^{c}$ for $k\geq 2$. Then $(B_{k})$ is a partition of $A_{\ast}$, and since $\cap_{i}V_{i}|B_{k}\neq\emptyset$ for all $k$, it follows from the stability of the $V_{i}$’s that $\cap_{i}V_{i}|A_{\ast}\neq\emptyset$. Fix $x_{0}\in X$, and modify each $V_{i}$ by $W_{i}=V_{i}|A_{\ast}+\{x_{0}\}|A_{\ast}^{c}$. Then $\emptyset\neq\cap_{i}W_{i}=:W$ is stable, and

[TABLE]

is the infimum of $(V_{i}|A_{i})$.

We provide the construction of the complement. For $V|A\in\mathcal{P}(X)$, define

[TABLE]

By completeness of the ordered set $(\mathcal{P}(X),\sqsubseteq)$, $(V|A)^{\sqsubset}$ is well-defined. ∎

The symbols $\sqcup,\sqcap$ and ⊏, defined in (2.3), (2.4) and (2.5), are called conditional union, conditional intersection and conditional complement, respectively. We give another description of the conditional complement:

[TABLE]

where

[TABLE]

and

[TABLE]

By an exhaustion argument (similarly to the construction of the conditional intersection in the previous proof), $B_{\ast}$ is attained. It can directly be verified that $Z$ is stable. To see the equality in (2.6), notice that $(Z|B_{\ast}+X|A^{c})\sqcap V|A=\{\ast\}$, and therefore $Z|B_{\ast}+X|A^{c}\sqsubseteq(V|A)^{\sqsubset}$. By way of contradiction, suppose that $(Z|B_{\ast}+X|A^{c})\sqcap(V|A)^{\sqsubset}\neq\{\ast\}$. We may assume that $B_{\ast}\in\mathscr{F}_{+}$. Since $(V|A)^{\sqsubset}|A^{c}=X|A^{c}$, there must exist $C\in\mathscr{F}_{+}$ with $C\subseteq B_{\ast}$ and $x\in X$ such that $x|C\in(V|A)^{\sqsubset}$ and $x|C^{\prime}\not\in V|C^{\prime}$ for all $C^{\prime}\subseteq C$ with $C^{\prime}\in\mathscr{F}_{+}$. However, this contradicts the maximality of $B_{\ast}$.

As a consequence of elementary results in the theory of Boolean algebras, see e.g. [22, p. 14], one has:

Corollary 2.7**.**

For a stable set $X$, the structure $(\mathcal{P}(X),\sqcup,\sqcap,{}^{\sqsubset},\{\ast\},X)$ is a complete Boolean algebra.

The importance of the previous result lies in the validity of all Boolean laws in $\mathcal{P}(X)$ which are known from the classical power set algebra. The Boolean laws are fundamental in topology and measure theory. The application of the Boolean laws is usually referred to as Boolean arithmetic, see e.g. [22]. Compared with the usual power set, the conditional power set has additionally the property that its elements can be concatenated which brings some new properties which are specific to the conditional power set and are summarized in the following proposition.

Proposition 2.8**.**

Let $X$ be a stable set, $V|A\in\mathcal{P}(X)$, $(V_{i}|A_{i})_{i\in I}$, $(V_{k}|A_{k})_{k\in\mathbb{N}}$ and $(V_{k,j}|A_{k,j})_{k\in\mathbb{N},j\in J}$ be families in $\mathcal{P}(X)$, where $I$ and $J$ are arbitrary nonempty index sets. Let $(D_{k})$ be a partition of $\Omega$. Then the following are true.

• (S1)

$\sqcap_{i}(V_{i}|A_{i})=(\sqcap_{i}V_{i})|\inf_{i}A_{i}$* and $\sqcup_{i}(V_{i}|A_{i})=(\sqcup_{i}V_{i})|\sup_{i}A_{i}$.*

• (S2)

$(\sum(V_{k}|A_{k})|D_{k})\sqcap V|A=\sum(V_{k}|A_{k}\sqcap V|A)|D_{k}$* and $(\sum(V_{k}|A_{k})|D_{k})\sqcup V|A=\sum(V_{k}|A_{k}\sqcup V|A)|D_{k}$.*

• (S3)

$\sum(\sqcap_{j}V_{k,j}|A_{k,j})|D_{k}=\sqcap_{j}\sum(V_{k,j}|A_{k,j})|D_{k}$* and $\sum(\sqcup_{j}V_{k,j}|A_{k,j})|D_{k}=\sqcup_{j}\sum(V_{k,j}|A_{k,j})|D_{k}$.*

• (S4)

$(\sum(V_{k}|A_{k})|D_{k})^{\sqsubset}=\sum(V_{k}|A_{k})^{\sqsubset}|D_{k}$.

Proof.

We prove (S1) and (S4); the remaining two claims can be shown similarly by using the definitions of the conditional set operations.

• (S1)

We show $\sqcap_{i}(V_{i}|A_{i})=(\sqcap_{i}V_{i})|\inf_{i}A_{i}$. Suppose $\sqcap_{i}(V_{i}|A_{i})=W|A_{\ast}$ and $\sqcap_{i}V_{i}=W^{\prime}|B_{\ast}$ according to (2.4), and put $C_{\ast}=B_{\ast}\cap\inf_{i}A_{i}$. If $x|A_{\ast}\in W|A_{\ast}$, then $x|A_{\ast}\in W_{i}|A_{\ast}$ for all $i$, and thus $x|A_{\ast}\in W^{\prime}|A_{\ast}\sqsubseteq W^{\prime}|C_{\ast}$ since $A_{\ast}\subseteq C_{\ast}$. If now $C_{\ast}\setminus A_{\ast}\in\mathscr{F}_{+}$, then this contradicts the maximality of $A_{\ast}$, which implies that $W^{\prime}|C_{\ast}\sqsubseteq W|A_{\ast}$.

We show $\sqcup_{i}(V_{i}|A_{i})=(\sqcup_{i}V_{i})|A$, where $A=\sup_{i}A_{i}$. Suppose $\sqcup_{i}(V_{i}|A_{i})=W|A$ and $\sqcup V_{i}=W^{\prime}$ according to $\eqref{eq:union}$. By stability, it follows that $W|A\subseteq W^{\prime}|A$. As for the converse, let $(A_{i_{k}})$ be a countable subfamily of $(A_{i})$ such that $\sup_{k}A_{i_{k}}=A$ and $\sum x_{h}|B_{h}\in W^{\prime}$ for some partition $(B_{h})$ of $\Omega$ and $x_{h}\in X_{i}$ for some $i$ and all $h$. Then $(B_{h}\cap A_{i_{k}})_{h,k}$ is a partition of $A$, and extending the family $(x_{h})$ to a family $(x_{h,k})$ by $x_{h,k}:=x_{h}$ for all $h,k$ we obtain $(\sum x_{h}|B_{h})|A=\sum x_{h,k}|B_{h}\in W|A$, which shows that $W^{\prime}|A\subseteq W|A$.

• (S4)

According to (2.6), let $(V_{k}|A_{k})^{\sqsubset}=U_{k}|B_{k,\ast}+X|A_{k}^{c}$ for each $k$, and

[TABLE]

where $A=\cup_{k}(A_{k}\cap D_{k})$ and the first equality follows from (2.2). On the one hand, we have

[TABLE]

On the other hand, $U|B_{\ast}\cap(A_{k}\cap D_{k})=U_{k}|B_{k,\ast}\cap D_{k}$ for all $k$, which implies

[TABLE]

∎

Remark 2.9**.**

In many concepts and proofs in conditional set theory, it is necessary to construct the largest set $A\in\mathscr{F}$ on which a property is satisfied. For example see the definition of the conditional intersection (2.4) and the representation of the conditional complement (2.6). This can be achieved by an exhaustion argument which requires that the considered property is stable under countable concatenations which follows from the stability of sets and relations. Without further notice, if we write “by an exhaustion argument”, then we mean that an exhaustion can is applied.

Finally, we recall the definition of a stable function, cf. [9, Definition 2.17].

Definition 2.10**.**

Let $X$ and $Y$ be stable sets. A function $f\colon X\to Y$ is said to be a stable function if $f(\sum x_{k}|A_{k})=\sum f(x_{k})|A_{k}$ for all partitions $(A_{k})$ and every sequence $(x_{k})$ in $X$. Let $W|A\sqsubseteq Y$ and put

[TABLE]

By an exhaustion argument, $C_{\ast}$ is attained. Let

[TABLE]

Define the conditional pre-image of $W|A$ as

[TABLE]

For a sequence of stable functions $f_{k}\colon X\to Y$ and a partition $(A_{k})$, define their concatenation by

[TABLE]

which is a stable function from $X$ to $Y$.

3. A conditional version of Carathéodory’s extension theorem and the Lebesgue integral

The aim of this section is to develop basic results of measure theory in a conditional context. Our focus lies on Carathéodory’s extension and uniqueness theorems and the construction of a Lebesgue integral.

Definition 3.1**.**

Let $X$ be a stable set of functions on $\Omega$. A stable collection $\mathcal{X}$ on $X$ is called a

• •

stable ring whenever $V|A\sqcap(W|B)^{\sqsubset},V|A\sqcup W|B\in\mathcal{X}$ for all $V|A,W|B\in\mathcal{X}$;

• •

stable Dynkin system whenever $X\in\mathcal{X}$, $(V|A)^{\sqsubset}\in\mathcal{X}$ for all $V|A\in\mathcal{X}$, and $\sqcup_{k}(V_{k}|A_{k})\in\mathcal{X}$ for all sequences $(V_{k}|A_{k})$ of pairwise disjoint222$V|A$ and $W|B$ are said to be disjoint if $V|A\sqcap W|B=\{\ast\}$. elements in $\mathcal{X}$;

• •

stable $\sigma$-algebra whenever $X\in\mathcal{X}$, $(V|A)^{\sqsubset}\in\mathcal{X}$ for all $V|A\in\mathcal{X}$, and $\sqcup_{k}(V_{k}|A_{k})\in\mathcal{X}$ for all sequences $(V_{k}|A_{k})$ in $\mathcal{X}$.

The pair $(X,\mathcal{X})$, where $\mathcal{X}$ is a stable $\sigma$-algebra, is called a stable measurable space. If $(Y,\mathcal{Y})$ is another stable measurable space, then a stable function $f\colon X\to Y$ is called stably measurable if $f^{-1}(V|A)\in\mathcal{X}$ for all $V|A\in\mathcal{Y}$.

Remark 3.2**.**

Since the intersection of a nonempty family of stable Dynkin systems (stable $\sigma$-algebras) is a stable Dynkin system (stable $\sigma$-algebra), we can define the stable Dynkin system (stable $\sigma$-algebra) generated by a collection $\mathcal{E}$ of conditional sets, henceforth denoted by $\mathcal{D}(\mathcal{E})$ ($\Sigma(\mathcal{E})$).

Examples 3.3**.**

Let $X$ be a stable set of functions on $\Omega$. Then $\{X|A\colon A\in\mathscr{F}\}$ and $\mathcal{P}(X)$ are stable $\sigma$-algebras, called the trivial and the discrete stable $\sigma$-algebra on $X$.

• 2)

Let $L^{0}(E)$ be the space of all strongly measurable functions with values in a Banach space $E$. The norm $\|\cdot\|$ of $E$ extends to an $L^{0}$-valued norm on $L^{0}(E)$ by defining $\|x\|(\omega):=\|x(\omega)\|$ a.s. Let $\mathcal{E}$ be the collection of all stable open balls $B_{r}(x):=\{y\in L^{0}(E)\colon\|x-y\|<r\}$, $x\in L^{0}(E)$ and $r\in L^{0}_{++}$. By straightforward inspection, $\mathcal{E}$ is a stable collection in $\mathcal{P}(L^{0}(E))$ which is a base for a stable topology on $L^{0}(E)$, see [9, Section 3]. We call the stable $\sigma$-algebra $\Sigma(\mathcal{E})$ the stable Borel $\sigma$-algebra on $L^{0}(E)$, and denote it by $\mathcal{B}(L^{0}(E))$.

The following proposition is an immediate consequence of Boolean arithmetic.

Proposition 3.4**.**

A stable Dynkin system is a stable $\sigma$-algebra if and only if it is closed under finite conditional intersections.

The previous results can be used to establish the following Dynkin’s $\pi$-$\lambda$ type result.

Theorem 3.5**.**

Let $\mathcal{E}$ be a stable collection which is closed under finite conditional intersections. Then $\Sigma(\mathcal{E})=\mathcal{D}(\mathcal{E})$.

Proof.

By Corollary 3.4, it is enough to prove that $\mathcal{D}(\mathcal{E})$ is closed under finite conditional intersections. For $V|A\in\mathcal{D}(\mathcal{E})$, let $\mathcal{F}_{V|A}:=\{W|B\in\mathcal{P}(X)\colon V|A\sqcap W|B\in\mathcal{D}(\mathcal{E})\}$. By (S3) and the stability of $\mathcal{D}(\mathcal{E})$, it follows that $\mathcal{F}_{V|A}$ is a stable collection. By Boolean arithmetic, it follows that $\mathcal{F}_{V|A}$ is a stable Dynkin system. For every $V|A\in\mathcal{E}$, one has $\mathcal{E}\subseteq\mathcal{F}_{V|A}$ from the assumption, and therefore $\mathcal{D}(\mathcal{E})\subseteq\mathcal{F}_{V|A}$. If now $V|A\in\mathcal{E}$ and $W|B\in\mathcal{D}(\mathcal{E})$, then $W|B\in\mathcal{F}_{V|A}$, and thus $V|A\in\mathcal{F}_{W|B}$. It follows that $\mathcal{E}\subseteq\mathcal{F}_{W|B}$ and $\mathcal{D}(\mathcal{E})\subseteq\mathcal{F}_{W|B}$ which shows the claim. ∎

Remark 3.6**.**

Let $(X,\mathcal{X})$ and $(Y,\mathcal{Y})$ be stable measurable spaces, $\mathcal{E}$ a collection in $\mathcal{P}(Y)$ generating $\mathcal{Y}$ and $f\colon X\to Y$ a stable function. Then $f$ is stably measurable if and only if $f^{-1}(V|A)\in\mathcal{X}$ for all $V|A\in\mathcal{E}$. Indeed, by stability of $f$, $\mathcal{P}(X)$ and $\mathcal{P}(Y)$, the collection

[TABLE]

is stable. One has $Y\in\mathcal{Z}$, and $\mathcal{Z}$ is closed under conditional complementation and countable conditional unions. Thus $\mathcal{Y}\subseteq\mathcal{Z}$ if and only if $\mathcal{E}\subseteq\mathcal{Z}$.

If $\mathcal{X}$ is a stable collection in $\mathcal{P}(X)$, then a function $\mu\colon\mathcal{X}\to\bar{L}^{0}_{+}$ is said to be a stable set function if

[TABLE]

for all sequences $(V_{k}|B_{k})$ in $\mathcal{X}$ and every partition $(A_{k})$, where the concatenation on the l.h.s is defined as in (2.2).

Definition 3.7**.**

Let $X$ be a stable set of functions on $\Omega$ and $\mathcal{X}$ a stable ring on $X$. A function $\mu:\mathcal{X}\to\bar{L}^{0}_{+}$ is called a stable pre-measure whenever $\mu$ is a stable function and

• (M1)

$\mu(V|A)=\mu(V|A)|A+0|A^{c}$ for all $V|A\in\mathcal{X}$;

• (M2)

$\mu(\sqcup_{k}(V_{k}|A_{k}))=\sum_{k\geq 1}\mu(V_{k}|A_{k})$ for all sequences $(V_{k}|A_{k})$ of pairwise disjoint elements in $\mathcal{P}(X)$ with $\sqcup_{k}(V_{k}|A_{k})\in\mathcal{X}$.333The infinite series in $\bar{L}^{0}_{+}$ is understood pointwise a.s.

If the domain of a stable pre-measure $\mu$ is a stable $\sigma$-algebra, then $\mu$ is called a stable measure and the triple $(X,\mathcal{X},\mu)$ a stable measure space. A stable measure $\mu$ is said to be

• •

finite if $\mu(X)<\infty$, and a stable probability measure if $\mu(X)=1$;

• •

$\sigma$-finite if there exists an increasing sequence $(V_{k}|A_{k})$ in $\mathcal{X}$ with $\sqcup_{k}(V_{k}|A_{k})=X$ such that $\mu(V_{k}|A_{k})<\infty$ for all $k$.

One can see property (M1) as the local analogue of $\mu(\emptyset)=0$ for classical measures. We give one simple example of a stable measure and will provide further examples in Section 4.

Example 3.8**.**

Let $(X,\mathcal{X})$ be a stable measurable space and $x\in X$. The stable Dirac measure centered at $x$ is defined by

[TABLE]

where $B=\sup\{B^{\prime}\in\mathscr{F}\colon B^{\prime}\subseteq A,x|B^{\prime}\in V|B^{\prime}\}$ which is attained by an exhaustion argument.

Similarly to the classical case, we have the following elementary properties of stable pre-measures.

Proposition 3.9**.**

Let $X$ be a stable set of functions on $\Omega$, $\mathcal{X}$ a stable ring and $\mu\colon\mathcal{X}\to\bar{L}^{0}_{+}$ a stable function satisfying property (M1) and finite additivity (i.e. (M2) for finite sequences). Then the following are true.

• (M3)

$\mu(V|A\sqcup W|B)+\mu(V|A\sqcap W|B)=\mu(V|A)+\mu(W|B)$* for all $V|A,W|B\in\mathcal{X}$;*

• (M4)

$\mu(V|A)\leq\mu(W|B)$* whenever $V|A\sqsubseteq W|B$;*

• (M5)

$\mu(W|B\sqcap(V|A)^{\sqsubset})=\mu(W|B)-\mu(V|A)$* whenever $V|A\sqsubseteq W|B$ and $\mu(V|A)<\infty$;*

• (M6)

if $\mu$ is a stable pre-measure, then $\mu(\sqcup_{k}V_{k}|A_{k})\leq\sum_{k}\mu(V_{k}|A_{k})$ for all sequences $(V_{k}|A_{k})$ in $\mathcal{X}$ with $\sqcup_{k}V_{k}|A_{k}\in\mathcal{X}$;

• (M7)

$\mu$* is a stable pre-measure if and only if $\mu(V_{k}|A_{k})\uparrow\mu(\sqcup_{k}V_{k}|A_{k})$ a.s. for all increasing sequences $(V_{k}|A_{k})$ in $\mathcal{X}$ with $\sqcup_{k}V_{k}|A_{k}\in\mathcal{X}$;*

• (M8)

if $\mu(V|A)<\infty$ for all $V|A\in\mathcal{X}$, then $\mu$ is a stable pre-measure if and only if $\mu(V_{k}|A_{k})\downarrow\mu(\sqcap_{k}V_{k}|A_{k})$ a.s. for all decreasing sequences $(V_{k}|A_{k})$ in $\mathcal{X}$ with $\sqcap_{k}V_{k}|A_{k}\in\mathcal{X}$.

Proof.

Since the verification of the statements is similar to the proof of the respective classical statements, we only provide the main arguments.

• (M3)

We have

[TABLE]

Let $C=\{\mu(W|B\sqcap(V|A)^{\sqsubset})<\infty\}$. On $C$ we get (M3) by adding the previous two identities and subtracting $\mu(W|B\sqcap(V|A)^{\sqsubset})$. On $C^{c}$ we have $\mu(V|A\sqcup W|B)=\mu(W|B)=\infty$ which also implies (M3).

• (M4)

and (M5) follow from finite additivity.

• (M6)

can be shown by finite additivity, (M4) and Boolean arithmetic.

• (M7)

follows from finite additivity and Boolean arithmetic.

• (M8)

follows from (M7) on applying (M5).

∎

Definition 3.10**.**

Let $X$ be a stable set of functions on $\Omega$. A stable function $\mu^{\ast}\colon\mathcal{P}(X)\to\bar{L}^{0}_{+}$ is called a stable outer measure whenever

• (A1)

$\mu^{\ast}((V|A)|B)=\mu^{\ast}(V|A)|B+0|B^{c}$ for all $V|A\in\mathcal{P}(X)$ and $B\in\mathscr{F}$;

• (A2)

$\mu^{\ast}(V|A)\leq\mu^{\ast}(W|B)$ whenever $V|A\sqsubseteq W|B$;

• (A3)

$\mu^{\ast}(\sqcup_{k}(V_{k}|A_{k}))\leq\sum_{k}\mu^{\ast}(V_{k}|A_{k})$ for all sequences $(V_{k}|A_{k})$ in $\mathcal{P}(X)$.

An element $V|A\in\mathcal{P}(X)$ is said to be stably $\mu^{\ast}$-measurable whenever

[TABLE]

A stable measure can be obtained from a stable outer measure as follows.

Proposition 3.11**.**

Let $X$ be a stable set of functions on $\Omega$ and $\mu^{\ast}\colon\mathcal{P}(X)\to\bar{L}^{0}_{+}$ a stable outer measure. Then $\mu^{\ast}$ is a stable measure on the stable collection $\mathcal{X}(\mu^{\ast})$ of all stably $\mu^{\ast}$-measurable sets.

Proof.

From the stability of $\mu^{\ast}$ and (S2), we obtain that $\mathcal{X}(\mu^{\ast})$ is a stable collection. Clearly, $X\in\mathcal{X}(\mu^{\ast})$, and by symmetry, $\mathcal{X}(\mu^{\ast})$ is closed under conditional complementation. Let $V|A,W|B\in\mathcal{X}(\mu^{\ast})$. Then

[TABLE]

Replacing $Z|C$ by $Z|C\sqcap V|A$ and $Z|C\sqcap(V|A)^{\sqsubset}$ in (3.2), respectively, and taking the sum of the two resulting equations, we obtain

[TABLE]

Replace $Z|C$ by $Z|C\sqcap(V|A\sqcup W|B)$ in (3), and get from Boolean arithmetic

[TABLE]

Plugging in (3.4) in (3), one has $V|A\sqcup W|B\in\mathcal{X}(\mu^{\ast})$. By de Morgan’s law, $\mathcal{X}(\mu^{\ast})$ is also closed under finite intersections. Let $(U_{k}|D_{k})$ be a sequence of pairwise disjoint sets in $\mathcal{X}(\mu^{\ast})$ and $U|D=\sqcup_{k}U_{k}|D_{k}$. Choosing $V|A=U_{1}|D_{1}$ and $W|B=U_{2}|D_{2}$ in (3.4), one gets

[TABLE]

By induction,

[TABLE]

From the previous we know that $Y_{n}|E_{n}=\sqcup_{k\leq n}(U_{k}|D_{k})\in\mathcal{X}(\mu^{\ast})$. By (A2), $Z|C\sqcap(U|D)^{\sqsubset}\sqsubseteq Z|C\sqcap(Y_{n}|E_{n})^{\sqsubset}$ implies $\mu^{\ast}(Z|C\sqcap(U|D)^{\sqsubset})\leq\mu^{\ast}(Z|C\sqcap(Y_{n}|E_{n})^{\sqsubset})$ for all $Z|C\in\mathcal{P}(X)$. Therefore, we obtain from (3.5)

[TABLE]

for all $Z|C\in\mathcal{P}(X)$ and $n\in\mathbb{N}$. By (A3),

[TABLE]

for all $Z|C\in\mathcal{P}(X)$, which means $U|D\in\mathcal{X}(\mu^{\ast})$. By Theorem 3.5, $\mathcal{X}(\mu^{\ast})$ is a stable $\sigma$-algebra. To see that $\mu^{\ast}:\mathcal{X}(\mu^{\ast})\to\bar{L}^{0}_{+}$ is a stable measure, replace $Z|C$ by $U|D$ in (3.6) and note that the reverse inequality follows from (A3). ∎

This leads us to a conditional version of Carathéodory’s extension theorem.

Theorem 3.12**.**

Let $X$ be a stable set of functions on $\Omega$, $\mathcal{X}$ be a stable ring on $X$ and $\mu:\mathcal{X}\to\bar{L}^{0}_{+}$ be a stable pre-measure. Then there exists a stable measure $\nu\colon\Sigma(\mathcal{X})\to\bar{L}^{0}_{+}$ which coincides with $\mu$ on $\mathcal{X}$.

Proof.

For each $V|A\in\mathcal{P}(X)$, let

[TABLE]

By an exhaustion argument, $B_{V|A}$ is attained. Let $\mathcal{U}(V|A)$ be the collection of all sequences $(U_{k}|C_{k})$ in $\mathcal{X}$ such that $(V|A)|B_{V|A}\sqsubseteq\sqcup_{k}U_{k}|C_{k}$. By (S3) and the stability of $\mathcal{X}$ it follows that $\mathcal{U}(V|A)$ is a stable collection. Moreover, one has $B_{\sum(V_{k}|A_{k})|D_{k}}=\cup_{k}(B_{V_{k}|A_{k}}\cap D_{k})$ for all sequences $(V_{k}|A_{k})$ in $\mathcal{P}(X)$ and partitions $(D_{k})$ of $\Omega$. Thus $\mu^{\ast}\colon\mathcal{P}(X)\to\bar{L}^{0}_{+}$ defined by

[TABLE]

is a well-defined stable set function. We want to show that $\mu^{\ast}$ is a stable outer measure. Properties (A1) and (A2) are easy to check. As for (A3), let $(V_{k}|A_{k})$ be a sequence in $\mathcal{P}(X)$. Clearly, $B:=\cup_{k}B_{V_{k}|A_{k}}=B_{\sqcup_{k}(V_{k}|A_{k})}$. Fix $\varepsilon>0$. For each $k$, let $(U_{k,n}|C_{k,n})\in\mathcal{U}(V_{k}|A_{k})$ be such that

[TABLE]

Then

[TABLE]

which proves (A3).

Let $V|A\in\mathcal{X}$. We want to show that $V|A\in\mathcal{X}(\mu^{\ast})$, that is,

[TABLE]

Let $Z|C\in\mathcal{P}(X)$. On $B_{Z|C}^{c}$ there is nothing to show. On the other hand, we have $B_{Z|C\sqcap V|A}\cup B_{Z|C\sqcap(V|A)^{\sqsubset}}\subseteq B_{Z|C}$. Moreover, for $(U_{k}|C_{k})\in\mathcal{U}(Z|C)$ it follows from (M2) that

[TABLE]

Hence (3.7) is also satisfied on $B_{Z|C}$. We have shown $\mathcal{X}\subseteq\mathcal{X}(\mu^{\ast})$. Now it follows from Proposition 3.11 that the restriction of $\mu^{\ast}$ to $\Sigma(\mathcal{X})$, denoted by $\nu$, is a stable measure. Since $B_{V|A}=\Omega$ for $V|A\in\mathcal{X}$, and by (M4) and (M6),

[TABLE]

for all $(U_{k}|C_{k})\in\mathcal{U}(V|A)$, it follows that $\nu$ coincides with $\mu$ on $\mathcal{X}$. ∎

As for the uniqueness of the previous extension, we have the following result.

Proposition 3.13**.**

Let $(X,\mathcal{X})$ be a stable measurable space and $\mathcal{E}$ a stable generator of $\mathcal{X}$ which is closed under finite conditional intersections. For two stable measures $\mu,\nu$ on $(X,\mathcal{X})$, suppose $\mu(V|A)=\nu(V|A)$ for all $V|A\in\mathcal{E}$ and there exists a sequence $(Z_{k}|C_{k})$ in $\mathcal{E}$ with $\sqcup_{k}Z_{k}|C_{k}=X$ and $\mu(Z_{k}|C_{k})=\nu(Z_{k}|C_{k})<\infty$ for all $k$. Then $\mu=\nu$.

Proof.

The proof follows from a monotone class argument in the present context. Indeed, for $V|A\in\mathcal{E}$ with $\mu(V|A)=\nu(V|A)<\infty$, let

[TABLE]

Then $X\in\mathcal{D}_{V|A}$. Stability of $\mathcal{D}_{V|A}$ follows from (S3). By (M5) and Boolean arithmetic, $\mathcal{D}_{V|A}$ is closed under complementation. By (M2), $\mathcal{D}_{V|A}$ contains the conditional union of every pairwise disjoint sequence of its elements. We have that $\mathcal{D}_{V|A}$ is a stable Dynkin system with $\mathcal{E}\subseteq\mathcal{D}_{V|A}$. By Theorem 3.5, $\mathcal{D}_{V|A}=\mathcal{X}$. Hence,

[TABLE]

for all $W|B\in\mathcal{X}$ and $V|A\in\mathcal{E}$ with $\mu(V|A)=\nu(V|A)$. Therefore,

[TABLE]

for all $k$, and the claim follows from (M2). ∎

Fix a stable measure space $(X,\mathcal{X},\mu)$ in the remainder of this section. We construct a conditional Lebesgue integral for stably measurable functions $f\colon X\to L^{0}$, where we endow $L^{0}$ with the stable Borel $\sigma$-algebra $\mathcal{B}(L^{0})$ defined in Example 3.3.2). Consider the following stable generators of $\mathcal{B}(L^{0})$. For $r\in L^{0}$, let

[TABLE]

Notice that $[r,\infty[^{\sqsubset}=]-\infty,r[$ and $]r,\infty[^{\sqsubset}=]-\infty,r]$. The collection of all of each type of these intervals is a stable collection in $\mathcal{P}(L^{0})$, and by Boolean arithmetic, a generator of $\mathcal{B}(L^{0})$.

The concatenation of a sequence of stably measurable functions $f_{k}\colon X\to L^{0}$ along a partition $(A_{k})$ of $\Omega$ is defined in (2.8), and it is a stably measurable function. The sum $f+g$ and product $f\cdot g$ of two stably measurable functions $f,g\colon X\to L^{0}$ is defined pointwise, and it can be checked that they are stably measurable functions. Further, we write $f\leq g$ whenever $f(x)\leq g(x)$ in $L^{0}$ for all $x\in X$. It follows that $\max\{f,g\}$ and $\min\{f,g\}$ are stably measurable. The convergence of a sequence of stably measurable functions $f_{k}\colon X\to L^{0}$ to a function $f\colon X\to L^{0}$ is defined by

[TABLE]

for all $x\in X$, if this limit exists, in which case $f$ is stably measurable.

We introduce stable indicator functions and stable elementary functions. Let $V|A\in\mathcal{P}(X)$, and for each $x\in X$ let

[TABLE]

By an exhaustion argument, $A_{x}$ is attained. Further, $A_{\sum x_{k}|A_{k}}=\cup_{k}(A_{x_{k}}\cap A_{k})$ for every sequence $(x_{k})$ in $X$ and each partition $(A_{k})$ of $\Omega$. Thus the function $1_{V|A}\colon X\to L^{0}$ defined by $x\mapsto 1|A_{x}+0|A_{x}^{c}$ is well-defined and stable, and called the stable indicator function of $V|A$. The following properties of stable indicator functions can be directly checked from the definition.

• (D1)

$1_{V|A}=1_{V}|A+0|A^{c}$ for all $V|A\sqsubseteq X$;

• (D2)

$1_{\sum(V_{k}|A_{k})|B_{k}}=\sum 1_{V_{k}|A_{k}}|B_{k}$ for all sequences $(V_{k}|A_{k})$ in $\mathcal{P}(X)$ and partitions $(B_{k})$ of $\Omega$;

• (D3)

$1_{\sqcup_{i}V_{k}|A_{k}}=\sum_{k}1_{V_{k}|A_{k}}$ for all sequences $(V_{k}|A_{k})$ of pairwise disjoint elements in $\mathcal{P}(X)$;

• (D4)

$1_{V|A}=1-1_{(V|A)^{\sqsubset}}$ for all $V|A\sqsubseteq X$.

Definition 3.14**.**

Let $(r_{k})_{k\leq n}$ be a finite family in $L^{0}_{+}$ and $(V_{k}|A_{k})_{k\leq n}$ a finite family of pairwise disjoint elements in $\mathcal{X}$ with $\sqcup_{k}V_{k}|A_{k}=X$. The function $\sum_{k\leq n}r_{k}1_{V_{k}|A_{k}}\colon X\to L^{0}_{+}$ is called a stable elementary function.

Without further notice, we identify $\mathbb{N}$ with a subset of $L^{0}_{s}(\mathbb{N})$ by the embedding $n\mapsto n1_{\Omega}$ and $\mathbb{R}$ with a subset of $L^{0}$ by the embedding $r\mapsto r1_{\Omega}$. Also $L^{0}_{s}(\mathbb{N})$ can be understood as a subset of $L^{0}$.

Remark 3.15**.**

We can define $r\cdot s$ and $r+s$ for arbitrary $r,s\in\bar{L}^{0}_{+}$ by considering the conventions $a\cdot\infty=\infty\cdot a=\infty$ for all $a\in\mathbb{R}_{++}$, $a+\infty=\infty+a=\infty$ for all $a\in\mathbb{R}_{+}$ and $0\cdot\infty=\infty\cdot 0=0$ in a pointwise a.s. sense.

Definition 3.16**.**

For a stable elementary function $f=\sum_{k\leq n}r_{k}1_{V_{k}|A_{k}}$, we define its stable Lebesgue integral as

[TABLE]

To extend the previous definition to stably measurable functions, we need the following two lemmas.

Lemma 3.17**.**

Let $f\colon X\to L^{0}_{+}$ be a stable elementary function and $(f_{n})$ an increasing sequence of stable elementary functions $f_{n}\colon X\to L^{0}_{+}$ such that $f\leq\sup_{n}f_{n}$. Then it holds $\int_{X}fd\mu\leq\sup_{n}\int_{X}f_{n}d\mu$.

Proof.

Suppose $f=\sum_{k\leq m}r_{k}1_{V_{k}|A_{k}}$. Fix $r\in L^{0}$ with $0<r<1$. For every $n$, let $W_{n}|B_{n}:=(f_{n}-rf)^{-1}([0,\infty[)$. Then $f_{n}\geq rf1_{W_{n}|B_{n}}$, and therefore

[TABLE]

for each $n$. Since $(f_{n})$ is an increasing sequence with $f\leq\sup_{n}f_{n}$, it follows that $(W_{n}|B_{n})$ is an increasing sequence with $X=\sqcup_{n}W_{n}|B_{n}$. Thus $(W_{n}|B_{n}\sqcap V_{k}|A_{k})$ is an increasing sequence with $V_{k}|A_{k}=\sqcup_{n}(W_{n}|B_{n}\sqcap V_{k}|A_{k})$ for each $k$. By (M7),

[TABLE]

Then from (3.9)

[TABLE]

Since $r\in L^{0}$ with $0<r<1$ is arbitrary, the claim follows. ∎

Lemma 3.18**.**

For every stably measurable function $f\colon X\to L^{0}_{+}$ there exists an increasing sequence $(f_{n})$ of stable elementary functions such that $\sup_{n}f_{n}(x)=f(x)$ for all $x\in X$.

Proof.

We denote $\{f<r\}:=f^{-1}(]-\infty,r[)$ and $\{r\leq f\}:=f^{-1}([r,\infty[)$, $r\in L^{0}$. For each $n\in\mathbb{N}$, let

[TABLE]

Define

[TABLE]

Fix $x\in X$ and $n\in\mathbb{N}$. For $k=0,\ldots,n2^{n}-1$, we have

[TABLE]

where both suprema are attained by an exhaustion argument. Let

[TABLE]

Since $(W_{k,n}|A_{k,n})_{0\leq k\leq n2^{n}}$ is a partition of $X$, $(A_{k})_{0\leq k\leq n2^{n}}$ is a partition of $\Omega$. Hence from $f_{n}(x)\leq f_{n+1}(x)$ on $A_{k}$ for all $0\leq k\leq n2^{n}$ it follows that $f_{n}(x)\leq f_{n+1}(x)$. Thus $(f_{n})$ is increasing. By construction, $\sup_{n}f_{n}=f$. ∎

Since $L^{0}$ is a vector lattice, every stably measurable function $f\colon X\to L^{0}$ can be written as the difference $f^{+}-f^{-}$ of the stably measurable functions $f^{+}:=\max\{f,0\}$ and $f^{-}:=\max\{-f(x),0\}$. It follows from the previous two lemmas that the following is well-defined.

Definition 3.19**.**

Let $f\colon X\to L^{0}_{+}$ be stably measurable. Define

[TABLE]

where $(f_{n})$ is an increasing sequence of stable elementary functions with $\sup_{n}f_{n}=f$. Let $f\colon X\to L^{0}$ be stably measurable. We say that $f$ is integrable whenever $\int_{X}f^{+}d\mu,\int_{X}f^{-}d\mu\in L^{0}$. In this case, we define its stable Lebesgue integral by

[TABLE]

The following properties of the stable integral can be verified directly:

• (I1)

$\int_{X}\sum f_{k}|A_{k}d\mu=\sum\int_{X}f_{k}d\mu|A_{k}$ for every sequence $(f_{k})$ of integrable functions and every partition $(A_{k})$ of $\Omega$.

• (I2)

$\int_{X}fd\mu\leq\int_{X}gd\mu$ for every pair of integrable functions $f,g$ with $f\leq g$.

• (I3)

$\int_{X}rf+gd\mu=r\int_{X}fd\mu+\int_{X}gd\mu$ for all integrable functions $f,g$ and every $r\in L^{0}$.

We have the following version of the monotone convergence theorem.

Theorem 3.20**.**

Let $(f_{n})$ be an increasing sequence of integrable functions $f_{n}\colon X\to L^{0}_{+}$. Then it holds

[TABLE]

In particular,

[TABLE]

Proof.

Let $f:=\sup_{n}f_{n}$. By (I2), it is enough to build a sequence $(h_{n})$ of stable elementary functions with $\sup_{n}h_{n}=f$ and $h_{n}\leq f_{n}$ for all $n$. By Lemma 3.18, for each $f_{n}$ there exists a sequence $(g_{m,n})$ of stable elementary functions such that $f_{n}=\sup_{m}g_{m,n}$. Then $h_{m}:=\max_{k\leq m}g_{m,k}$ satisfies the required. ∎

4. Kernels, conditional distributions and stable measures

In this section, we establish a link between kernels and stable measures. Based thereupon, we can extend the representation of conditional distribution of random variables with values in a not necessarily standard Borel space by replacing probability kernels with stable probability measures. We apply this representation to compute the conditional expectation of functions of random variables by means of the stable Lebesgue integral.

Unless mentioned otherwise, $(E,\mathscr{E})$ denotes an arbitrary measurable space throughout this section. Recall that $L^{0}_{m}(E)$ denotes the space of all measurable functions $x\colon\Omega\to E$, which are called random variables with values in $E$. Let $\mathscr{R}$ be a classical ring of sets generating the $\sigma$-algebra $\mathscr{E}$. For every sequence $(F_{k})$ in $\mathscr{R}$ and each partition $(A_{k})$, let

[TABLE]

By definition, $\sum L^{0}_{m}(F_{k})|A_{k}$ is a stable subset of $L^{0}_{m}(E)$. Since $\mathscr{R}$ is a ring, by a straightforward computation,

[TABLE]

is a stable ring on $L^{0}_{m}(E)$. We consider the stable $\sigma$-algebra $\Sigma(\mathcal{R})$ generated by $\mathcal{R}$. If we replace in (4.2) the ring $\mathscr{R}$ by the larger $\sigma$-algebra $\mathscr{E}$, then the stable $\sigma$-algebra generated by the stable ring with respect to $\mathscr{E}$ coincides with the one generated by the stable ring with respect to $\mathscr{R}$. Indeed, this follows from the observation

[TABLE]

for all $(F_{n})$ and $F$ in $\mathscr{E}$. In particular, if $E$ is a separable topological space, by Boolean arithmetic, the stable measurable space $(L^{0}_{m}(E),\Sigma(\mathcal{R}))$ is identical with the stable Borel space $(L^{0}(E),\mathcal{B}(L^{0}(E)))$, see Examples 3.3.2) and Remark 2.3.

Recall that a kernel on $E$ is a function $\kappa\colon\Omega\times\mathscr{E}\to\overline{\mathbb{R}}_{+}$ such that $\kappa(\omega,F)$ is $\mathscr{F}$-measurable in $\omega\in\Omega$ for fixed $F\in\mathscr{E}$ and a measure in $F\in\mathscr{E}$ for fixed $\omega\in\Omega$. A kernel $\kappa$ is said to be a probability kernel if $\kappa(\omega,F)$ is a probability measure in $F\in\mathscr{E}$ for all $\omega\in\Omega$. We have the following main result which connects kernels with stable measures.

Theorem 4.1**.**

• (i)

For every kernel $\kappa\colon\Omega\times\mathscr{E}\to\overline{\mathbb{R}}_{+}$ there exists a stable measure on $(L^{0}_{m}(E),\Sigma(\mathcal{R}))$, denoted by $\mu_{\kappa}$, such that

[TABLE]

If $\kappa$ is a probability kernel, then $\mu_{\kappa}$ is the unique stable probability measure satisfying the previous equation.

• (ii)

Suppose $(E,\mathscr{E})$ is a standard Borel space. For every stable probability measure $\mu$ on the stable Borel space $(L^{0}(E),\mathcal{B}(L^{0}(E)))$ there exists a probability kernel on $E$, denoted by $\kappa_{\mu}$, such that

[TABLE]

In particular, one has the following reciprocality identities:

[TABLE]

Proof.

• (i)

For $(\sum L^{0}_{m}(F_{k})|A_{k})|A\in\mathcal{R}$, define

[TABLE]

If $\mu_{\kappa}$ is a stable pre-measure, then the first claim follows from Theorem 3.12. (M1) is satisfied by definition. As for (M2), let $((\sum L^{0}_{m}(F_{k,n})|A_{k,n})|A_{n})$ be a sequence of pairwise disjoint elements in $\mathcal{R}$ such that $\sqcup_{n}(\sum L^{0}_{m}(F_{k,n})|A_{k,n})|A_{n}\in\mathcal{R}$. Denoting $A:=\cup_{n}A_{n}$, the conditional union must be of the form

[TABLE]

Since the above sequence is pairwise disjoint, each $F_{k}$ is the disjoint union of all $F_{k,n}$ with $\emptyset\neq A\cap A_{k}\subseteq A_{n}\cap A_{k,n}$. Thus (M2) follows from the pointwise $\sigma$-additivity of the kernel $\kappa$. The second claim follows immediately from Proposition 3.13.

• (ii)

The proof can be carried out similarly to the one of the existence of regular conditional distributions. We will follow the main arguments in [21, Theorem 6.3].

By [21, Theorem A1.2], there exists a Borel isomorphism from $E$ to a Borel subset $S$ of $\mathbb{R}$. Therefore, it is enough to prove the claim for the Borel space $(S,\mathscr{B}(S))$. For each $q\in\mathbb{Q}$, let $f_{q}=f(\cdot,q)\colon\Omega\to[0,1]$ be defined by

[TABLE]

By (M4), one has $f(\cdot,p)\leq f(\cdot,q)$ whenever $p\leq q$. Let $A$ be the set of all $\omega\in\Omega$ such that $f(\omega,q)$ is increasing in $q\in\mathbb{Q}$ with limits $1$ and [math] at $\pm\infty$. Since $A$ is specified by countably many measurable conditions each of which holds a.s., we have $A\in\mathscr{F}$. Define

[TABLE]

From (M4) and (M8) it follows that $F(\omega,\cdot)$ is a distribution function for every $\omega\in\Omega$. Hence, by [21, Proposition 2.14], there exists a probability measure $\kappa(\omega,\cdot)$ such that $\kappa(\omega,]-\infty,x])=F(\omega,x)$ for all $x\in\mathbb{R}$ and $\omega\in\Omega$. By a monotone class argument, $\kappa$ is a probability kernel. Using a monotone class argument based on an a.s. interpretation of (M5) and (M7) shows that $\kappa(\cdot,F)=\mu(L^{0}_{m}(F))$ for all $F\in\mathscr{B}(\mathbb{R})$. In particular, $\kappa(\cdot,S^{c})=0$ a.s., and thus $\kappa_{\mu}(\cdot,F)=\mu(L^{0}_{m}(F))$ for all $F\in\mathscr{B}(S)$, where $\kappa_{\mu}$ is the probability kernel defined by

[TABLE]

where $s\in S$ is arbitrary. Uniqueness follows from (4.4) by a monotone class argument. The reciprocality identities are an immediate consequence of the constructions of $\kappa_{\mu}$ and $\mu_{\kappa}$.

∎

The first part of the previous theorem provides a procedure to construct stable measures from classical ones as follows.

Examples 4.2**.**

Let $\lambda$ be a $\sigma$-finite measure on $(E,\mathscr{E})$, and view $\lambda$ as a constant kernel on $\Omega\times\mathscr{E}$. Let $\mu_{\lambda}$ be the induced stable measure on $(L^{0}_{m}(E),\Sigma(\mathcal{R}))$ as in Theorem 4.1. Let $(F_{n})$ be an increasing sequence in $\mathscr{E}$ such that $\cup_{n}F_{n}=E$ and $\lambda(F_{n})<\infty$ for all $n$. Then $\sqcup_{n}L^{0}_{m}(F_{n})=L^{0}(E)$ and $\mu_{\lambda}(L^{0}_{m}(F_{n}))<\infty$ for all $n$, which implies that $\mu_{\lambda}$ is a $\sigma$-finite stable measure. By Proposition 3.13, $\mu_{\lambda}$ is the unique stable measure extending the classical measure $\lambda$ from $(E,\mathscr{E})$ to $(L^{0}_{m}(E),\Sigma(\mathcal{R}))$. In particular, if $(\mathbb{R}^{d},\mathscr{B}(\mathbb{R}^{d}),\lambda)$ is the classical Borel space with $\lambda$ the $d$-dimensional Lebesgue measure, then the extension $\mu_{\lambda}$ to the stable Borel space $((L^{0})^{d},\mathcal{B}((L^{0})^{d}))$ is called the stable $d$-dimensional Lebesgue measure.

Next, we study the link to the conditional distribution of random variables. For the remainder of this section, let $\mathscr{G}\subseteq\mathscr{F}$ be a sub-$\sigma$-algebra and $\xi:\Omega\to E$ a random variable. Recall that a regular conditional distribution of $\xi$ given $\mathscr{G}$ is a version of the function $\mathbb{P}[\xi\in\cdot|\mathscr{G}]$ on $\Omega\times\mathscr{E}$ which is a probability kernel. It is well known that such a representing probability kernel exists whenever $(E,\mathscr{E})$ is a standard Borel space, see e.g. [21, Theorem 6.3]. In view of Theorem 4.1, we can extend this representation result to arbitrary spaces as follows.

Example 4.3**.**

All objects will be defined relative to the probability space $(\Omega,\mathscr{G},\mathbb{P})$, e.g. $\mathscr{G}$-stable sets, $\mathscr{G}$-stable conditional power sets, $\mathscr{G}$-stable conditional set operations, etc. Let $L^{0}_{m,\mathscr{G}}(E)$ denote the space of all $\mathscr{G}$-measurable functions $x\colon\Omega\to E$. For a sequence $(F_{k})$ in $\mathscr{E}$, a partition $(A_{k})$ with $A_{k}\in\mathscr{G}$ for all $k$, define

[TABLE]

Let $\mathcal{R}_{\mathscr{G}}$ be the $\mathscr{G}$-stable ring on $L^{0}_{m,\mathscr{G}}(E)$ consisting of all $(\sum L^{0}_{m,\mathscr{G}}(F_{k})|A_{k})|A$, where $(F_{k})$ is a sequence in $\mathscr{E}$, $(A_{k})$ is a partition with $A_{k}\in\mathscr{G}$ for all $k$ and $A\in\mathscr{G}$. We denote by $\Sigma_{\mathscr{G}}(\mathcal{R}_{\mathscr{G}})$ the smallest $\mathscr{G}$-stable $\sigma$-algebra on $L^{0}_{m,\mathscr{G}}(E)$ including $\mathcal{R}_{\mathscr{G}}$. Then there exists a unique $\mathscr{G}$-stable probability measure on $(L^{0}_{m,\mathscr{G}}(E),\Sigma_{\mathscr{G}}(\mathcal{R}_{\mathscr{G}}))$, denoted by $\nu=\nu_{\xi,\mathscr{G}}$, such that

[TABLE]

Indeed, put

[TABLE]

and proceed similarly to the first part of the proof of Theorem 4.1.

We want to compute the conditional expectation of $f(\xi)$ with the help of the stable Lebesgue integral on applying the representation (4.5). Fix a Borel measurable function $f\colon E\to\mathbb{R}$ such that $\mathbb{E}[|f(\xi)|]<\infty$. Recall that the distribution of $\xi$ is defined by the pushforward measure $\mathbb{P}_{\xi}:=\mathbb{P}\circ\xi^{-1}$ on $(E,\mathscr{E})$. By the transformation theorem, it holds

[TABLE]

If $(E,\mathscr{E})$ is a standard Borel space, then we have the following conditional analogue of (4.6):

[TABLE]

where $\kappa_{\xi}$ is a regular conditional distribution of $\xi$ given $\mathscr{G}$. We can extend the representation (4.7) to spaces which are not necessarily standard Borel in the following way.

Theorem 4.4**.**

Let $f\colon E\to\mathbb{R}$ be a Borel measurable function with $\mathbb{E}[|f(\xi)|]<\infty$. Then $f$ extends to a $\mathscr{G}$-stable integrable function $\hat{f}\colon L^{0}_{m,\mathscr{G}}(E)\to L^{0}_{\mathscr{G}}$ such that

[TABLE]

Proof.

By inspection, $\hat{f}\colon L^{0}_{m,\mathscr{G}}(E)\to L^{0}_{\mathscr{G}}$ given by $\hat{f}(x):=f\circ x$ is well-defined and $\mathscr{G}$-stable. We show that $\hat{f}$ satisfies (4.8). We can assume w.l.o.g. that $f\geq 0$. First, suppose $f=\sum_{k\leq n}x_{k}1_{F_{k}}$. Then $\hat{f}$ is equal to the $\mathscr{G}$-stable elementary function $\sum_{k\leq n}x_{k}1_{L^{0}_{m,\mathscr{G}}(F_{k})}$. It follows from (I3) and (4.5) that

[TABLE]

Second, suppose $f\geq 0$. Let $(f_{n})$ be an increasing sequence of simple functions such that $f=\lim f_{n}$. Then $(\hat{f}_{n})$ is an increasing sequence of stable elementary functions such that $\hat{f}$ is the pointwise limit of $(\hat{f}_{n})$. By the monotone convergence theorem for conditional expectations and Theorem 3.20, we obtain from (4.9)

[TABLE]

∎

5. A conditional version of Fubini’s theorem, the Radon-Nikodým theorem, the Daniell-Stone theorem and the Riesz representation theorem

In this section, we establish four important theorems in measure theory for stable measure spaces. We start with Fubini’s theorem and the Radon-Nikodým theorem, and close with the Daniell-Stone theorem from which we derive two Riesz type represenation and regularity results.

5.1. Fubini’s theorem.

Throughout this subsection, let $(X,\mathcal{X},\mu)$ and $(Y,\mathcal{Y},\nu)$ be two stable probability spaces. On the stable set $X\times Y$ of functions on $\Omega$ consider the conditional rectangles $V|A\times W|B:=V\times W|A\cap B$ where $V|A\in\mathcal{X}$ and $W|B\in\mathcal{Y}$. Inspection shows

[TABLE]

which implies that the collection

[TABLE]

is a stable collection in $\mathcal{P}(X\times Y)$ closed under finite conditional intersections. We define the stable product $\sigma$-algebra of $\mathcal{X}$ and $\mathcal{Y}$ to be the stable $\sigma$-algebra generated by $\mathcal{E}$, and denote it by $\mathcal{X}\otimes\mathcal{Y}$. Let $V|A\sqsubseteq X\times Y$ and $x\in X$. The conditional $x$-section of $V|A$ is defined to be the conditional set

[TABLE]

where

[TABLE]

Since $D_{\ast}$ is attained by an exhaustion argument, $W$ is a stable subset of $Y$ by stability of $V$. Similarly, we define the conditional $y$-section of $V|A$ for some $y\in Y$. For a function $f\colon X\times Y\to E$, we denote by $f_{x}$ and $f_{y}$ its $x$- and $y$-section, respectively, which are defined classically as $f_{x}(y)=f_{y}(x)=f(x,y)$. If $f$ is a stable function, then $f_{x}$ and $f_{y}$ are stable functions as well. We have the following useful properties which we state for (conditional) $x$-sections; analogous properties hold for $y$-sections.

Proposition 5.2**.**

Let $Z|C\in\mathcal{P}(X\times Y)$, $(Z_{i}|C_{i})_{i\in I}$ and $(Z_{k}|A_{k})_{k\in\mathbb{N}}$ be families in $\mathcal{P}(X\times Y)$, where $I$ is an arbitrary nonempty index set. Let $f\colon X\times Y\to L^{0}$ be a function. Let $x\in X$, $(x_{k})$ be a sequence in $X$ and $(D_{k})$ a partition. Then the following holds true.

• (F1)

$(\sum(Z_{k}|A_{k})|D_{k})_{x}=\sum(Z_{k}|A_{k})_{x}|D_{k}$;

• (F2)

$(Z|C)_{\sum x_{k}|D_{k}}=\sum(Z|C)_{x_{k}}|D_{k}$;

• (F3)

$(\sqcup_{i}Z_{i}|C_{i})_{x}=\sqcup_{i}(Z_{i}|C_{i})_{x}$;

• (F4)

$((Z|C)_{x})^{\sqsubset}=((Z|C)^{\sqsubset})_{x}$;

• (F5)

if $Z|C\in\mathcal{X}\otimes\mathcal{Y}$, then $(Z|C)_{x}\in\mathcal{Y}$;

• (F6)

if $f$ is conditionally $\mathcal{X}\otimes\mathcal{Y}$-measurable, then $f_{x}$ is conditionally $\mathcal{Y}$-measurable.

• (F7)

for $Z|C\in\mathcal{X}\otimes\mathcal{Y}$, the function $x\mapsto\nu((Z|C)_{x})$ is conditionally $\mathcal{X}$-measurable.

Proof.

The statements (F1) and (F2) can easily be verified from the definitions.

• (F3)

Suppose $\sqcup_{i}Z_{i}|C_{i}=Z|C$ where $C=\sup_{i}C_{i}$. Let

[TABLE]

By definition of the conditional union (see (2.3)), one has $D_{\ast}\subseteq\sup_{i}D^{i}_{\ast}$. To see the converse inclusion, let $(D_{i_{k}})$ be such that $\sup_{k}D_{i_{k}}=\sup_{i}D_{i}$. Let $(y_{i_{k}})$ be a sequence in $Y$ such that $(x,y_{i_{k}})|D_{i_{k}}\in Z_{i_{k}}|D_{i_{k}}$ for all $k$. Let $C_{1}=D_{i_{1}}$ and $C_{k}=C_{i_{k}}\cap(C_{1}\cup\ldots\cup C_{i_{k-1}})^{c}$ for $k\geq 2$. For $y=\sum y_{i_{k}}|C_{k}+y_{0}|(\sup_{k}D_{i_{k}})^{c}$, where $y_{0}\in Y$ is arbitrary, one has $(x,y)|\sup_{k}D_{i_{k}}\in Z|\sup_{k}D_{i_{k}}$, which shows $\sup_{i}D_{i}=\sup_{k}D_{i_{k}}\subseteq D$. We conclude that $(Z|C)_{x}=\sqcup_{i}(Z_{i}|C_{i})_{x}$.

• (F4)

By (F3),

[TABLE]

Suppose $(Z|C)^{\sqsubset}=W|B$. Let

[TABLE]

From (5.3) it follows $D^{1}_{\ast}\cup D^{2}_{\ast}=\Omega$. Also, it holds $D_{1}\cap D_{2}=\emptyset$ since otherwise $Z|C\sqcap(Z|C)^{\sqsubset}\neq\{\ast\}$ which contradicts the complementation law in the Boolean algebra $\mathcal{P}(X)$. We conclude that the conditional union $Y=(Z|C)_{x}\sqcup((Z|C)^{\sqsubset})_{x}$ is a disjoint conditional union, and this implies $((Z|C)_{x})^{\sqsubset}=((Z|C)^{\sqsubset})_{x}$.

• (F5)

Let $\mathcal{Z}$ be the collection of all $Z|C\in\mathcal{X}\otimes\mathcal{Y}$ such that $(Z|C)_{x}\in\mathcal{Y}$. Then $\mathcal{Z}$ includes all conditional rectangles $V|A\times W|B$ where $V|A\in\mathcal{X}$ and $W|B\in\mathcal{Y}$. By (F3) and (F4), $\mathcal{Z}$ is closed under conditional complementation and countable conditional unions. Since $\mathcal{Y}$ is a stable collection, we conclude that $\mathcal{Z}$ is a stable $\sigma$-algebra. Since $\mathcal{Z}$ includes the stable generator $\mathcal{E}$ (see (5.1)) and $\mathcal{X}\otimes\mathcal{Y}$ is the smallest stable $\sigma$-algebra including $\mathcal{E}$, it follows that $\mathcal{Z}=\mathcal{X}\otimes\mathcal{Y}$.

• (F6)

By (F5), $f^{-1}_{x}(]r,\infty[)=(f^{-1}(]r,\infty[))_{x}\in\mathcal{Y}$ for all $r\in L^{0}$.

• (F7)

By (F2) and the stability of $\nu$, the function $x\mapsto\nu((Z|C)_{x})$ is stable. Let

[TABLE]

From $X\times Y\in\mathcal{D}$ and (F1) it follows that $\mathcal{D}$ is a stable collection. From (F3) and (M2), it follows that $\mathcal{D}$ is closed under taking countable conditional unions of pairwise disjoint conditional sets. From (F4) and (M5), $\mathcal{D}$ is also closed under conditional complementation. Thus $\mathcal{D}$ is a stable Dynkin system. Since for $V|A\in\mathcal{X}$ and $W|B\in\mathcal{Y}$ the function $x\mapsto\nu((V|A\times W|B)_{x})=\nu(W|B)1_{V|A}(x)$ is stably $\mathcal{X}$-measurable, $\mathcal{D}=\mathcal{X}\otimes\mathcal{Y}$ due to Theorem 3.5.

∎

Lemma 5.3**.**

There exists a unique stable probability measure $\lambda$ on $\mathcal{X}\otimes\mathcal{Y}$ such that

[TABLE]

for all $V|A\in\mathcal{X}$ and $W|B\in\mathcal{Y}$. Moreover, for all $Z|C\in\mathcal{X}\otimes\mathcal{Y}$, one has

[TABLE]

where $s^{X}_{Z|C}(x):=\nu((Z|C)_{x})$, $x\in X$, and $s^{Y}_{Z|C}(y):=\mu((Z|C)_{y})$, $y\in Y$.

We call $\mu\otimes\nu:=\lambda$ the stable product measure of $\mu$ and $\nu$ and the triple $(X\times Y,\mathcal{X}\otimes\mathcal{Y},\mu\otimes\nu)$ the stable product probability space.

Proof.

For $Z|C\in\mathcal{X}\otimes\mathcal{Y}$, define

[TABLE]

By (F2), the stability of $\nu$ and (I1), $\lambda$ is a stable function on $\mathcal{X}\otimes\mathcal{Y}$ satisfying (M1). That $\lambda$ satisfies (M2) follows from (F3) and Theorem 3.20. For $V|A\in\mathcal{X}$ and $W|B\in\mathcal{Y}$, one has

[TABLE]

By analogous arguments, the mapping

[TABLE]

is a stable probability measure with the above property. Since $\mathcal{E}$ as defined in (5.1) is a stable generator of $\mathcal{X}\otimes\mathcal{Y}$ closed under finite conditional intersections, the claim follows from Proposition 3.13. ∎

We have the following version of Fubini’s theorem, which can easily be extended to $\sigma$-finite stable measure spaces.

Theorem 5.4**.**

Let $f\colon X\times Y\to L^{0}$ be a stable integrable function with respect to $(X\times Y,\mathcal{X}\otimes\mathcal{Y},\mu\otimes\nu)$. Then the functions

[TABLE]

are stable integrable functions, and one has

[TABLE]

Proof.

First, suppose that $f$ is a stable elementary function of the form $\sum_{k\leq n}r_{k}1_{Z_{k}|C_{k}}$. Then $f_{x}=\sum_{k\leq n}r_{k}1_{(Z_{k}|C_{k})_{x}}$ is stably measurable due to (F6). By (F7),

[TABLE]

is stably measurable and $L^{0}$-bounded. By Lemma 5.3,

[TABLE]

Second, let $f$ be a non-negative stable integrable function, and choose with the help of Lemma 3.18 an increasing sequence $(f_{n})$ of stable elementary functions such that $f=\sup_{n}f_{n}$. By Theorem 3.20, (I2) and the first step,

[TABLE]

Finally, for an arbitrary stable integrable function $f$, the previous equality follows from the identities $|f|_{x}=(f_{+})_{x}+(f_{-})_{x}$ and $f_{x}=(f_{+})_{x}-(f_{-})_{x}$, (I3) and the first and second step. Analogously, one can prove

[TABLE]

which finishes the proof. ∎

Remark 5.5**.**

A stable function $K\colon X\times\mathcal{Y}\to L^{0}_{+}$ is said to be a stable Markov kernel if

• (i)

$K(x,\cdot)$ is a stable probability measure for all $x\in X$,

• (ii)

$K(\cdot,V|A)$ is stably measurable for all $V|A\in\mathcal{Y}$.

We can extend Fubini’s theorem to stable Markov kernels. We provide the statement of this extension in the following, but will omit a proof as it can be worked out with a similar strategy as above. Let $\mu$ be a stable probability measure on $(X,\mathcal{X})$. Then $K\otimes\mu(V|A):=\int_{X}K(x,(V|A)_{x})d\mu$ defines a stable probability measure on $\mathcal{X}\otimes\mathcal{Y}$. If $f\colon X\times Y\to L^{0}$ is stably integrable w.r.t. $K\otimes\mu$, then

[TABLE]

5.6. Radon-Nikodým theorem.

Throughout this subsection, fix a stable measurable space $(X,\mathcal{X})$, and let $\mu$ be a stable probability measure on $(X,\mathcal{X})$. If $f\geq 0$ is a stable integrable function with $\int_{X}fd\mu=1$, then from (D1)-(D3) and Theorem 3.20 we have that

[TABLE]

is a stable probability measure absolutely continuous w.r.t. $\mu$, where a stable probability measure $\nu$ is said to be absolutely continuous w.r.t. $\mu$ whenever $V|A\in\mathcal{X}$ and $\mu(V|A)=0$ imply $\nu(V|A)=0$. In this subsection, we prove the following converse statement.

Theorem 5.7**.**

If $\nu$ is a stable probability measure absolutely continuous relative to $\mu$, then there exists a stable integrable function $f\colon X\to L^{0}$ such that

[TABLE]

for all $V|A\in\mathcal{X}$

By a straightforward extension, a Radon-Nikodým theorem can also be established in the case where $\mu$ and $\nu$ are $\sigma$-finite. The proof is based on the following auxiliary result.

Lemma 5.8**.**

Let $\mu_{1}$ and $\mu_{2}$ be finite stable measures on $(X,\mathcal{X})$ and define $\mu_{3}:=\mu_{2}-\mu_{1}$. Then there exists $X_{0}|A_{0}\in\mathcal{X}$ such that

• (i)

$\mu_{3}(X)\leq\mu_{3}(X_{0}|A_{0})$;

• (ii)

$0\leq\mu_{3}(V|A)$* for all $V|A\in\mathcal{X}$ with $V|A\sqsubseteq X_{0}|A_{0}$.*

Proof.

First, we establish the weaker statement: For all $r\in L^{0}_{++}$ there is $X_{r}|A_{r}\in\mathcal{X}$ such that

• $(i)^{\prime}$

$\mu_{3}(X)\leq\mu_{3}(X_{r}|A_{r})$;

• $(ii)^{\prime}$

$-r<\mu_{3}(V|A)$ for all $V|A\in\mathcal{X}$ with $V|A\sqsubseteq X_{r}|A_{r}$.

Let $B_{0}:=\{\mu_{3}(X)\leq 0\}$. If $B_{0}=\Omega$, then $X_{r}|A_{r}:=\{\ast\}$ satisfies (i)′ and (ii)′. Otherwise $B_{0}^{c}\in\mathscr{F}_{+}$. In this case let

[TABLE]

By an exhaustion argument, $B_{1}$ is attained. If $B_{1}=B_{0}^{c}$, then $X_{r}|A_{r}:=X|B_{1}$ fulfills the required. Indeed, by stability and (M1),

[TABLE]

which shows (i)′, and (ii)′ follows from the definition of $B_{1}$. If $B_{0}^{c}\cap B_{1}^{c}\in\mathscr{F}_{+}$, then by the maximality of $B_{1}$ there exists $V_{1}|A_{1}\in\mathcal{X}$ with $V_{1}|A_{1}\sqsubseteq X|(B_{0}\cup B_{1})^{c}$ such that $\mu_{3}(V_{1}|A_{1})|(B_{0}\cup B_{1})^{c}\leq-r|(B_{0}\cup B_{1})^{c}$. Thus

[TABLE]

So if

[TABLE]

is equal to $(B_{0}\cup B_{1})^{c}$, then

[TABLE]

satisfies (i)′ and (ii)′. Otherwise we have $B_{0}^{c}\cap B_{1}^{c}\cap B_{2}^{c}\in\mathscr{F}_{+}$ and there exists $V_{2}|A_{2}\in\mathcal{X}$ with $V_{2}|A_{2}\sqsubseteq(V_{1}|A_{1})^{\sqsubset}|(B_{0}\cup B_{1}\cup B_{2})^{c}$ such that $\mu_{3}(V_{2}|A_{2})|(B_{0}\cup B_{1}\cup B_{2})^{c}\leq-r|(B_{0}\cup B_{1}\cup B_{2})^{c}$. Since $V_{1}|A_{1}\sqcap V_{2}|A_{2}=\{\ast\}$, one has similarly to the above computation

[TABLE]

and we can repeat the previous procedure. If this procedure does not yield the desired after finitely many steps, we obtain a sequence $(B_{n})_{n\geq 0}$ of pairwise disjoint elements in $\mathscr{F}$ and a sequence $(V_{n}|A_{n})_{n\geq 1}$ of pairwise disjoint sets in $\mathcal{X}$ such that

[TABLE]

and

[TABLE]

for all $n\geq 1$. If $B:=\cap_{n\geq 0}B_{n}^{c}\in\mathscr{F}_{+}$, then by (M2),

[TABLE]

which contradicts the finiteness of $\mu_{3}$. Thus $(B_{n})_{n\geq 0}$ is a partition of $\Omega$ and $X_{r}|A_{r}:=\sum W_{n}|B_{n}$ where $W_{0}:=\{\ast\}$, $W_{1}:=X$ and $W_{n}:=(V_{1}|A_{1}\sqcup\ldots\sqcup V_{n-1}|A_{n-1})^{\sqsubset}$, $n\geq 2$, satisfies (i)′ and (ii)′.

Second, we apply the previously established weaker statement to prove the claim. For every $n\in\mathbb{N}$, we can recursively choose $X_{1/n}|A_{1/n}$ such that $\mu_{3}(X)\leq\mu_{3}(X_{1/n}|A_{1/n})$ and $-1/n<\mu_{3}(V|A)$ for all $V|A\in\mathcal{X}$ with $V|A\sqsubseteq X_{1/n}|A_{1/n}$ and such that $X_{1/(n+1)}|A_{1/(n+1)}\sqsubseteq X_{1/n}|A_{1/n}$. Then $X_{0}|A_{0}:=\sqcap_{n}X_{1/n}|A_{1/n}$ fulfills (i)′ and (ii)′ due to (M8). ∎

We prove Theorem 5.7.

Proof.

Let $\mathcal{H}$ be the collection of all stable integrable functions $f\colon X\to L^{0}_{+}$ such that

[TABLE]

for all $V|A\in\mathcal{X}$. Inspection shows that $\mathcal{H}$ is upward directed. Let $r=\sup_{f\in\mathcal{H}}\int_{X}fd\mu\leq 1$. There exists an increasing sequence $(f_{k})$ in $\mathcal{H}$ such that $\sup_{k}\int_{X}f_{k}d\mu=r$. By Theorem 3.20, we have $\int_{X}fd\mu=r$ for $f:=\sup_{k}f_{k}$. By another application of Theorem 3.20, we see that

[TABLE]

It remains to show that $\lambda:=\nu-\tilde{\nu}=0$. By contradiction, suppose that $C:=\{\lambda(X)>0\}\in\mathscr{F}_{+}$. W.l.o.g. assume $C=\Omega$. Let $s:=\lambda(X)/2$. Applying Lemma 5.8 to $\mu_{2}=\lambda$ and $\mu_{1}=s\mu$ supplies us with $X_{0}|A_{0}\in\mathcal{X}$ such that $\lambda(X_{0}|A_{0})-s\mu(X_{0}|A_{0})\geq\lambda(X)-s\mu(X)>0$ and $\lambda(V|A)\geq s\mu(V|A)$ for all $V|A\in\mathcal{X}$ with $V|A\sqsubseteq X_{0}|A_{0}$. It can be checked that $\tilde{f}:=f+s1_{X_{0}|A_{0}}\in\mathcal{H}$ with $\int_{X}\tilde{f}d\mu>r$ which is a contradiction. ∎

5.9. Daniell-Stone theorem and Riesz representation theorem.

The Daniell-Stone theorem and the Riesz representation theorem provide sufficient conditions under which a positive linear functional on a vector lattice is an integral. In this last section, we establish a conditional version of both theorems which state when a positive $L^{0}$-linear function on a stable vector lattice is a stable integral. The proof of the Daniell-Stone theorem in the present setting is an adaptation of a proof of the classical statement by using Carathéodory’s extension theorem, and therefore we omit its proof. We prove Riesz type representation and regularity results for positive $L^{0}$-linear functions on the stable vector lattice of all stable sequentially continuous functions $f\colon(L^{0})^{d}\to L^{0}$ and its stable subspace consisting of functions with stably compact conditional support, respectively.

Consider on $(L^{0})^{d}=L^{0}(\mathbb{R}^{d})$ the $L^{0}$-valued Euclidean norm, see Example 2.2.5), and endow it with the stable Borel $\sigma$-algebra $\mathcal{B}^{d}:=\mathcal{B}((L^{0})^{d})$, see Example 3.3.2). A conditional subset $V|A\sqsubseteq(L^{0})^{d}$ is said to be

• •

$L^{0}$-bounded if there exists $M\in L^{0}_{+}$ such that $\|x|A\|=\|x\||A\leq M|A$ for all $x|A\in V|A$;

• •

sequentially closed if $V|A$ includes the limit of every a.s. convergent sequence in $V|A$;

• •

stably compact if $V|A$ is $L^{0}$-bounded and sequentially closed.444Stable compactness refers to the interpretation of conditional compactness in a classical context. The characterization, which is stated above, is based on a conditional version of the Heine-Borel theorem, see [9, Theorem 4.6].

Let $X$ be a stable set of functions on $\Omega$. Recall that for a sequence $(f_{k})$ of stable functions $f_{k}\colon X\to L^{0}$ and a partition $(A_{k})$, we define $\sum f_{k}|A_{k}\colon X\to L^{0}$ by $(\sum f_{k}|A_{k})(x):=\sum f_{k}(x)|A_{k}$, $x\in X$. We call a space $\mathcal{L}$ of functions $f\colon X\to L^{0}$ stable if $\mathcal{L}\neq\emptyset$ and $\sum f_{k}|A_{k}\in\mathcal{L}$ for all sequences $(f_{k})$ in $\mathcal{L}$ and every partition $(A_{k})$ of $\Omega$.

Definition 5.10**.**

Let $X$ be a stable set of functions on $\Omega$. A set $\mathcal{L}$ of stable functions $f\colon X\to L^{0}$ is said to be a stable Stone vector lattice if $\mathcal{L}$ is stable and $f+rg,\min\{f,g\},\min\{f,1\}\in\mathcal{L}$ for all $f,g\in\mathcal{L}$ and $r\in L^{0}$.555Here, $1$ denotes the function with the constant value $1=1_{\Omega}\in L^{0}$.

Examples 5.11**.**

Let $\text{cl}(V|A)$ denote the sequential closure of $V|A$. Let $f\colon(L^{0})^{d}\to L^{0}$ be a stable function. Its conditional support is defined by

[TABLE]

and $f$ is said to have stably compact conditional support if $\text{supp}(f)$ is stably compact. A function $f\colon(L^{0})^{d}\to L^{0}$ is said to be sequentially continuous if $f(x_{n})\to f(x)$ a.s. whenever $x_{n}\to x$ a.s. Let $\mathcal{C}$ denote the space of all stable and sequentially continuous functions, and $\mathcal{C}_{c}$ denote its subspace of functions with stably compact conditional support. Then both $\mathcal{C}$ and $\mathcal{C}_{c}$ are stable Stone vector lattices.

Given a stable Stone vector lattice $\mathcal{L}$, a function $L\colon\mathcal{L}\to L^{0}$ is said to be

• •

stable if $L(\sum f_{k}|A_{k})=\sum L(f_{k})|A_{k}$ for all sequences $(f_{k})$ in $\mathcal{L}$ and every partition $(A_{k})$;

• •

$L^{0}$-linear if $L(f+rg)=L(f)+rL(g)$ for all $f,g\in\mathcal{L}$ and $r\in L^{0}$;

• •

continuous from above if $L(f_{n})\downarrow 0$ a.s. whenever $f_{n}(x)\downarrow 0$ a.s. for all $x\in X$.

We have the following conditional version of the Daniell-Stone theorem.

Theorem 5.12**.**

Let $\mathcal{L}$ be a stable Stone vector lattice and $L:\mathcal{L}\to L^{0}$ stable, $L^{0}$-linear, continuous from above and such that $L(f)\geq 0$ whenever $f\geq 0$. Then there exists a stable measure $\mu$ on

[TABLE]

such that

[TABLE]

In the following proofs of the conditional Riesz representation results it is necessary to pass from the sequential continuity of a sequence of stable functions to its sequential uniform continuity. This can be achieved by the following conditional version of Dini’s theorem. In the statement of the next lemma, we use the following extension of a sequence of stable functions. If $(f_{n})$ is a classical sequence of stable functions, we can extend $(f_{n})$ to a stable net parametrized by $L^{0}_{s}(\mathbb{N})$ by defining $f_{n}:=\sum f_{n_{k}}|A_{k}$ for $n=\sum n_{k}|A_{k}\in L^{0}_{s}(\mathbb{N})$.

Lemma 5.13**.**

Let $X\subset(L^{0})^{d}$ be stably compact and $(f_{n})$ a decreasing sequence of stable and sequentially continuous functions $f_{n}\colon X\to L^{0}$. Let $f\colon X\to L^{0}$ be stable sequentially continuous such that $f_{n}(x)\to f(x)$ a.s. for all $x\in X$. Then, for every $r\in L^{0}_{++}$ there exists $n_{0}\in L^{0}_{s}(\mathbb{N})$ such that $\sup_{x\in X}|f_{n}(x)-f(x)|<r$ for all $n\geq n_{0}$.

Proof.

The statement can be proved similarly to the respective classical statement, see e.g. [1, Theorem 2.66], by using the characterization of stable compactness in terms of stable open coverings, see [9]. ∎

We introduce the following regularity conditions for stable measures on $(L^{0})^{d}$.

Definition 5.14**.**

A stable measure $\mu$ on $\mathcal{B}^{d}$ is called

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conditionally closed regular whenever

[TABLE]

for all $V|A\in\mathcal{B}^{d}$;

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conditionally regular whenever

[TABLE]

for all $V|A\in\mathcal{B}^{d}$.

Theorem 5.15**.**

Let $L:\mathcal{C}\to L^{0}$ be stable, $L^{0}$-linear and such that $L(f)\geq 0$ whenever $f\geq 0$. Then there exists a conditionally regular finite stable measure $\mu$ on $\mathcal{B}^{d}$ such that

[TABLE]

Proof.

Let $(f_{l})$ be a sequence of stable functions in $\mathcal{C}$ such that $f_{l}\downarrow 0$. Let $V_{k}:=\{x\in(L^{0})^{d}\colon\|x\|\leq k\}$ for $k\in\mathbb{N}$, and notice that $V_{k}$ is stably compact. By triangle inequality, $d(\cdot,V_{k}):=\inf\{\|\cdot-y\|\colon y\in V_{k}\}$ is sequentially continuous, and by [6, Theorem 4.4], the infimum is attained since it is also stable. Define $g_{k}:=\max\{1-d(\cdot,V_{k}),0\}$ and $h_{k,l}:=g_{k}f_{l}+(1-g_{k})f_{1}\|\cdot\|/2k$ for all $k,l\in\mathbb{N}$. By definition, $f_{l}\leq h_{k,l}$ for all $k,l$. Fix $r\in L^{0}_{++}$. By changing, if necessary, to a stable net, let $k\in L^{0}_{s}(\mathbb{N})$ be large enough such that $1/(2k)L(f_{1}(1-g_{k})\|\cdot\|)<r/2$. By Lemma 5.13, choose $l\in L^{0}_{s}(\mathbb{N})$ sufficiently large such that $L(g_{k}f_{l})<r/2$. We obtain

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By Theorem 5.12, there exists a finite stable measure $\mu$ on $\mathcal{B}^{d}$ such that

[TABLE]

As for the conditional tightness of $\mu$, let

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It follows from (M5) that $(L^{0})^{d}\in\mathcal{G}$, and thus from (S4) and the stability of $\mu$ that $\mathcal{G}$ is a stable collection. By definition, $\mathcal{G}$ is closed under conditional complementation. Let $(W_{n}|B_{n})$ be a sequence in $\mathcal{G}$. Fix $r\in L^{0}_{++}$. Let $Z_{n}|C_{n}\sqsubseteq W_{n}|B_{n}$ be sequentially closed with $\mu(W_{n}|B_{n}\sqcap(Z_{n}|C_{n})^{\sqsubset})<r/2^{n}$ for each $n$. By (M6) and Boolean arithmetic, we obtain $\mu((\sqcup_{n}W_{n}|B_{n})\sqcap(\sqcup_{n}(Z_{n}|C_{n})^{\sqsubset}))<r$. Since $r\in L^{0}_{++}$ is arbitrary, we have $\sqcup_{n}W_{n}|B_{n}\in\mathcal{G}$. It follows from (M5) that also $(\sqcup_{n}W_{n}|B_{n})^{\sqsubset}\in\mathcal{G}$. It remains to show that every stable open ball is conditionally closed regular. Let $B_{r}(x)$ be a stable open ball. Define $U_{n}:=\{x\in(L^{0})^{d}\colon d(x,B_{r}(x)^{\sqsubset})\geq 1/n\}$. Since $(U_{n})$ is a sequence of stable sequentially closed sets with $\sqcup_{n}U_{n}=B_{r}(x)$, it follows from (M7) that $B_{r}(x)$ is conditionally closed regular. Finally, for every $V|A\in\mathcal{B}^{d}$ we have $\mu(V|A\sqcap V_{n})\uparrow\mu(V|A)$ due to (M7), where $V_{n}:=\{x\in(L^{0})^{d}\colon\|x\|\leq n\}$ for $n\in\mathbb{N}$. Moreover, for every $n$ there exists a sequentially closed set $W_{n}|B_{n}\sqsubseteq V|A\sqcap V_{n}$ such that $\mu(W_{n}|B_{n})\geq\mu(V|A\sqcap V_{n})-1/n$. Since a stable sequentially closed subset of a stably compact set is stably compact (see [9, Proposition 3.27]), it follows that $\mu$ is conditionally tight. ∎

Theorem 5.16**.**

Let $L:\mathcal{C}_{c}\to L^{0}$ be stable, $L^{0}$-linear and such that $L(f)\geq 0$ whenever $f\geq 0$. Then there exists a stable measure $\mu$ on $\mathcal{B}^{d}$ such that

[TABLE]

Moreover, it holds

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$\mu([x,y])<\infty$* for all stably compact intervals $[x,y]:=\{z\in(L^{0})^{d}\colon x\leq z\leq y\}$;666We consider on $(L^{0})^{d}$ the order $(x_{1},\ldots,x_{d})\leq(y_{1},\ldots,y_{d})$ if and only if $x_{i}\leq y_{i}$ for all $i=1,\ldots,d$.*

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$\mu(V|A)=\sup\{\mu(W|B)\colon W|B\sqsubseteq V|A\text{ stably compact}\}$* for all $V|A\in\mathcal{B}^{d}$ with $\mu(V|A)<\infty$.*

Proof.

In order to obtain the assumptions of Theorem 5.12, given a sequence $(f_{n})$ of stable functions in $\mathcal{C}_{c}$ such that $f_{n}\downarrow 0$, apply Lemma 5.13 to the sequence $(1_{\text{supp}(f_{1})}f_{n})$. We have $\mu([x,y])\leq\int_{(L^{0})^{d}}fd\mu=L(f)<\infty$, where $f=\max\{1-d(\cdot,[x,y]),0\}$. The regularity condition can be shown similarly to Theorem 5.15. ∎