Approximate Monge solutions continuously depending on the parameter

TL;DR

This paper studies how approximate optimal transportation solutions vary continuously with a parameter, ensuring stability of solutions in parametric optimal transport problems.

Contribution

It proves the existence of parameter-dependent approximate Monge mappings that are continuous, extending the understanding of stability in optimal transport.

Findings

Existence of continuous approximate Monge mappings.

Stability results for optimal transport solutions under parameter variation.

Extension to costs and marginals depending continuously on a parameter.

Abstract

We consider Kantorovich optimal transportation problem in the case where the cost function and marginal distributions continuously depend on a parameter with values in a metric space. We prove the existence of approximate optimal Monge mappings continuous with respect to the parameter.

Equations

$$K_{h}(\mu,\nu)=\inf\Bigl{\{}\int h\,d\sigma:\sigma\in\Pi(\mu,\nu)\Bigr{\}}$$

$$M_{h}(\mu,\nu)=\inf\Bigl{\{}\int h(x,T(x))\,\mu(dx):\mu\circ T^{-1}=\nu\Bigr{\}}$$

$$\mu\mapsto\biggl{|}\int f\,d\mu\biggr{|},$$

$$d((x_1,y_1),(x_2,y_2))=d_X(x_1,x_2)+d_Y(y_1,y_2).$$

$$d_{KR}(\mu,\nu)=\sup\biggl{\{}\int f\,d(\mu-\nu)\colon f\in{\rm Lip}_{1},\ |f|\leq 1\biggr{\}},$$

$$\int hd\sigma \le K_h(\mu ,\nu )+\epsilon .$$

$$h_{t}(x,y)\leq a_{t}(x)+b_{t}(y),\quad\lim\limits_{R\to+\infty}\sup_{t}\biggl{(}\int_{\{a_{t}\geq R\}}a_{t}\,d\mu_{t}+\int_{\{b_{t}\geq R\}}b_{t}\,d\nu_{t}\biggr{)}=0.$$

$$\lim_{R\to+\infty}\sup_{t\in T}\Bigl{(}\int_{a_{t}\geq R}a_{t}d\mu+\int_{b_{t}\geq R}b_{t}d\nu\Bigr{)}=0.$$

$$0<\mu (X\setminus K_1)=\mu (X\setminus {\tilde{K}}_1)+\lambda ([0,\mu ({\tilde{K}}_1)]\setminus S)<{\epsilon }_1.$$

$$\delta (t)=\sum _{\alpha }{\kappa }_{\tau (\alpha )}{\psi }_{\alpha }(t).$$

$$S=\underset{j=1}{\overset{\infty }{⨆}}{S}_j(t)$$

$$S_j(t)=S\cap [(j-1)\tilde{\delta }(t),j\tilde{\delta }(t)),j\in \mathbb{N}.$$

$${\int }_{X\times Y}g(x,y)I_{X_j(t)}{\pi }_{t_n}(dxdy)\to {\int }_{X\times Y}g(x,y)I_{X_j(t)}{\pi }_t(dxdy).$$

$${\int }_{X\times Y}f(x)g(x,y){\pi }_{t_n}(dxdy)\to {\int }_{X\times Y}f(x)g(x,y){\pi }_t(dxdy),$$

$$\Bigl{|}\int_{X\times Y}(I_{X_{j}(t)}g(x,y)\pi_{t_{n}}(dxdy)-\int_{X\times Y}f(x)g(x,y)\pi_{t_{n}}(dxdy)\Bigr{|}\leq\\ \leq\int_{X\times Y}|(I_{X_{j}(t)}-f(x))g(x,y)|\pi_{t_{n}}(dxdy)\leq\int_{X\times Y}I_{U_{j}\setminus F_{j}}\pi_{t_{n}}(dxdy)=\mu(U_{j}\setminus F_{j})<\delta.$$

$$\Bigl{|}\int_{X\times Y}g(x,y)I_{X_{j}(t)}\pi_{t_{n}}(dxdy)-\int_{X\times Y}g(x,y)I_{X_{j}(t)}\pi_{t}(dxdy)\Bigr{|}\leq\\ \leq\Bigl{|}\int_{X\times Y}f(x)g(x,y)\pi_{t_{n}}(dxdy)-\int_{X\times Y}f(x)g(x,y)\pi_{t}(dxdy)\Bigr{|}+2\delta.$$

$$\lambda {|}_{[0,\lambda (S_j(t))]}\circ {\xi }_{t,j}^{-1}={\nu }_t^j$$

$$F_t^j(s)=\lambda ([0,s]\cap S_j(t)).$$

$$T_t(x)={\xi }_{t,j}(F_t^j({\phi }^{-1}(x)))\text{mbox}ifx\in X_j(t),j\in \mathbb{N}.$$

$$\mu {|}_{X\setminus K_1}\circ T^{-1}=\nu -\alpha \nu {|}_{K_2}.$$

$$\mu (\{x\in X_j(t):T_{t_n}(x)\nrightarrow T_t(x)\})=0.$$

$$T_{t_n}(x)={\xi }_{t_n,j}(F_{t_n}^j({\phi }^{-1}(x)))\to {\xi }_{t,j}(F_t^j({\phi }^{-1}(x)))=T_t(x),$$

$$\Bigl{|}\int_{X_{j}(t)}h_{t}(x,T_{t}x)\mu(dx)-\int_{K_{2}}h_{t}(x_{0},y)\nu^{j}_{t}(dy)\Bigr{|}=\\ =\Bigl{|}\int_{X_{j}(t)}(h_{t}(x,T_{t}x)-h_{t}(x_{0},T_{t}x))\mu(dx)\Bigr{|}<\varepsilon_{1}\mu(X_{j}(t)),$$

$$\Bigl{|}\int_{X_{j}(t)\times K_{2}}h_{t}(x,y)\pi_{t}(dxdy)-\int_{K_{2}}h_{t}(x_{0},y)\nu^{j}_{t}(dy)\Bigr{|}=\\ =\Bigl{|}\int_{X_{j}(t)\times K_{2}}(h_{t}(x,y)-h_{t}(x_{0},y))\pi_{t}(dxdy)\Bigr{|}<\varepsilon_{1}\mu(X_{j}(t)).$$

$${\int }_{X_j(t)}{h}_t(x,T_{t}x)\mu (dx)\le {\int }_{X_j(t)\times K_2}{h}_t(x,y){\pi }_t(dxdy)+2{\epsilon }_1\mu (X_j(t)).$$

$${\int }_{K_1}{h}_t(x,T_{t}x)\mu (dx)\le {\int }_{K_1\times K_2}{h}_t(x,y){\pi }_t(dxdy)+2{\epsilon }_1.$$

$${\int }_X{h}_t(x,T_{t}x)\mu (dx)\le {\int }_{K_1\times K_2}{h}_t(x,y){\pi }_t(dxdy)+3{\epsilon }_1.$$

$$\begin{gathered} {\int }_{K_1\times K_2}{h}_t(x,y){\pi }_t(dxdy)\le {\int }_{K_1\times K_2}{h}_t(x,y)\tilde{\sigma }(dxdy)+{\epsilon }_1\le \\ \le {\int }_{K_1\times K_2}{h}_t(x,y)\sigma (dxdy)+(\nu (K_2)-{\nu }_1(K_2))+{\epsilon }_1. \end{gathered}$$

$${\int }_X{h}_t(x,T_{t}x)\mu (dx)\le {\int }_{K_1\times K_2}{h}_t(x,y){\pi }_t(dxdy)+3{\epsilon }_1\le {\int }_{X\times Y}{h}_t(x,y)\sigma (dxdy)+5{\epsilon }_1.$$

$$\begin{gathered} \int h_{t}d\sigma -\int \min (h_t,N)d\sigma \le \int h_t{I}_{\{h_t\ge N\}}d\sigma \le \\ \le \int (2a_t{I}_{\{a_t\ge N/2\}}+2b_t{I}_{\{b_t\ge N/2\}})d\sigma =2{\int }_{a_t\ge N/2}{a}_{t}d\mu +2{\int }_{b_t\ge N/2}{b}_{t}d\nu . \end{gathered}$$

Full text

Approximate Monge solutions continuously depending on the parameter

S.N. Popova 111Moscow Institute of Physics and Technology; National Research University Higher School of Economics.

Abstract. We consider Kantorovich optimal transportation problem in the case where the cost function and marginal distributions continuously depend on a parameter with values in a metric space. We prove the existence of approximate optimal Monge mappings continuous with respect to the parameter.

Keywords: optimal transportation problem, Kantorovich problem, Monge problem, continuity with respect to a parameter.

1. Introduction

We recall that, given two Borel probability measures $\mu$ and $\nu$ on topological spaces $X$ and $Y$ respectively and a nonnegative Borel function $h$ on $X\times Y$, the Kantorovich optimal transportation problem concerns minimization of the integral

[TABLE]

over all measures $\sigma$ in the set $\Pi(\mu,\nu)$ consisting of Borel probability measures on $X\times Y$ with projections $\mu$ and $\nu$ on the factors, that is, $\sigma(A\times Y)=\mu(A)$ and $\sigma(X\times B)=\nu(B)$ for all Borel sets $A\subset X$ and $B\subset Y$. The measures $\mu$ and $\nu$ are called marginal distributions or marginals, and $h$ is called a cost function. In general, there is only infimum $K_{h}(\mu,\nu)$, which may be infinite. If the cost function $h$ is continuous (or at least lower semicontinuous) and bounded and the measures $\mu$ and $\nu$ are Radon, then the minimum is attained and measures on which it is attained are called optimal measures or optimal Kantorovich plans. The boundedness of $h$ can be replaced by the assumption that there is a measure in $\Pi(\mu,\nu)$ with respect to which $h$ is integrable. The Monge problem for the same triple $(\mu,\nu,h)$ consists in finding a Borel mapping $T\colon X\to Y$ taking $\mu$ into $\nu$, that is $\nu=\mu\circ T^{-1}$, $(\mu\circ T^{-1})(B)=\mu(T^{-1}(B))$ for all Borel sets $B\subset Y$, for which the integral

[TABLE]

is minimal. In general, there is only infimum $M_{h}(\mu,\nu)$ (possibly, infinite), but in many interesting cases there exist optimal Monge mappings. In any case, $K_{h}(\mu,\nu)\leq M_{h}(\mu,\nu)$, but if both measures are Radon, $\mu$ has no atoms and is separable, and the cost function $h$ is continuous, then $K_{h}(\mu,\nu)=M_{h}(\mu,\nu)$ (see [9], [20]). This equality implies that if there is a unique solution $T$ to the Monge problem, then the image of $\mu$ under the mapping $x\mapsto(x,T(x))$ is an optimal Kantorovich plan. General information about Monge and Kantorovich problems can be found in [1], [10], [21], [22], and [24].

We consider optimal transportation of measures on metric and topological spaces in the case where the cost function $h_{t}$ and marginal distributions $\mu_{t}$ and $\nu_{t}$ depend on a parameter $t$ with values in a metric space. Kantorovich problems depending on a parameter were investigated in [24], [25], [18], [11], where the questions of measurability were studied. We address the problem of continuity with respect to the parameter. Here the questions naturally arise about the continuity with respect to $t$ of the optimal cost $K_{h_{t}}(\mu_{t},\nu_{t})$ and also about the possibility to select an optimal plan in $\Pi(\mu_{t},\nu_{t})$ continuous with respect to the parameter. In [12], [13] it was proved that the cost of optimal transportation is continuous with respect to the parameter in the case of continuous dependence of the cost function and marginal distributions on this parameter. Furthermore, it was shown that it is not always possible to select an optimal plan continuously depending on the parameter $t$. However, it is possible to select approximate optimal plans continuous with respect to the parameter. Continuous dependence on marginals was considered in [4], [23], and [16]. Similar problems may be studied for nonlinear cost functionals (see [17], [2], [3], [14], [19]), see also the recent survey [8].

Introduce the notation and terminology that will be used in this paper. A nonnegative Radon measure on a topological space $X$ is a bounded Borel measure $\mu\geq 0$ such that for every Borel set $B$ and every $\varepsilon>0$ there is a compact set $K\subset B$ such that $\mu(B\backslash K)<\varepsilon$ (see [5]). If $X$ is a complete separable metric space, then all Borel measures are Radon.

The space $\mathcal{M}_{r}(X)$ of signed bounded Radon measures on $X$ can be equipped with the weak topology generated by the seminorms

[TABLE]

where $f$ is a bounded continuous function.

A set $\mathcal{M}$ of nonnegative Radon measures on a space $X$ is called uniformly tight, if for every $\varepsilon>0$ there exists a compact set $K\subset X$ such that $\mu(X\backslash K)<\varepsilon$ for all $\mu\in\mathcal{M}$.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. The space $X\times Y$ is equipped with the metric

[TABLE]

The weak topology on the spaces of Radon probability measures $\mathcal{P}_{r}(X)$, $\mathcal{P}_{r}(Y)$, $\mathcal{P}_{r}(X\times Y)$ is metrizable by the corresponding Kantorovich–Rubinshtein metrics $d_{KR}$ (also called the Fortet–Mourier metrics, see [6]) defined by

[TABLE]

where ${\rm Lip}_{1}$ is the space of $1$-Lipschitz functions. If $X$ is complete, then $(\mathcal{P}_{r}(X),d_{KR})$ is also complete and if $X$ is Polish, then $\mathcal{P}_{r}(X)$ is also Polish.

In this paper we study the existence of approximate optimal Monge mappings continuous with respect to the parameter. Section 2 addresses the case where the measures $\mu\in\mathcal{P}_{r}(X)$ and $\nu\in\mathcal{P}_{r}(Y)$ are fixed and $h\colon X\times Y\times T\to[0,\infty)$ is a continuous cost function. In Section 3 we assume that the measure $\mu\in\mathcal{P}_{r}(X)$ is fixed and the measures $\nu_{t}\in\mathcal{P}_{r}(Y)$ continuously depend on $t$ in the weak topology. We prove that there exist approximate Monge solutions $T_{t}^{\varepsilon}$ such that $T_{t}^{\varepsilon}$ is continuous in $t$ in the sense of convergence $\mu$-a.e.: if $t_{n}\to t$ as $n\to\infty$, then $T_{t_{n}}^{\varepsilon}\to T_{t}^{\varepsilon}$ $\mu$-a.e. We also generalize this result to the case where the measures $\mu_{t}$ are continuous in $t$ in the total variation norm and the measures $\nu_{t}$ are continuous in $t$ in the weak topology.

2. The Monge problem with fixed marginals

In [12] the question was addressed whether it is possible to select an optimal plan continuously depending on the parameter $t$. The examples were constructed which show that such a choice is not always possible. However, the situation improves for approximate optimal plans. Given $\varepsilon>0$, a measure $\sigma\in\Pi(\mu,\nu)$ will be called $\varepsilon$-optimal for the cost function $h$ if

[TABLE]

Theorem 2.1** ([12]).**

Let $X$, $Y$ be complete metric spaces. Let $T$ be a metric space, and for every $t\in T$ we are given measures $\mu_{t}\in\mathcal{P}_{r}(X)$ and $\nu_{t}\in\mathcal{P}_{r}(Y)$ such that the mappings $t\mapsto\mu_{t}$ and $t\mapsto\nu_{t}$ are continuous in the weak topology (which is equivalent to the continuity in the Kantorovich–Rubinshtein metric). Suppose also that there is a continuous nonnegative function $(t,x,y)\mapsto h_{t}(x,y)$. Suppose that for every $t$ there exist nonnegative Borel functions $a_{t}\in L^{1}(\mu_{t})$ and $b_{t}\in L^{1}(\nu_{t})$ such that

[TABLE]

Then one can select $\varepsilon$-optimal measures $\sigma_{t}^{\varepsilon}\in\Pi(\mu_{t},\nu_{t})$ for the cost functions $h_{t}$ such that they will be continuous in $t$ in the weak topology for every fixed $\varepsilon>0$.

If for every $t$ there is a unique optimal plan $\sigma_{t}$, then it is continuous in $t$.

In this paper we strengthen the result from [12] looking at approximate optimal Monge mappings continuously depending on the parameter.

First, we consider the particular case where the marginals $\mu\in\mathcal{P}_{r}(X)$, $\nu\in\mathcal{P}_{r}(Y)$ are fixed and cost functions $h_{t}$ depend on the parameter $t$. We prove the following result on the existence of approximate optimal Monge mappings continuously depending on the parameter $t$.

Theorem 2.2**.**

Let $X,Y$ be completely regular topological spaces. Let $\mu$ be a non-atomic Radon probability measure on $X$, let $\nu$ be a Radon probability measure on $Y$, and the measures $\mu$ and $\nu$ are concentrated on countable unions of metrizable compact sets (i.e. we may assume that $X$ and $Y$ are Souslin spaces). Let $T$ be a metric space, $h\colon X\times Y\times T\to[0,\infty)$ be a continuous function such that $h(x,y,t)\leq a_{t}(x)+b_{t}(y)$, where $a_{t}\in L^{1}(\mu)$, $b_{t}\in L^{1}(\nu)$ and

[TABLE]

Then for any $\varepsilon>0$ one can select $\varepsilon$-optimal Monge mappings $T_{t}^{\varepsilon}$ for the cost functions $h_{t}$ such that $T_{t}^{\varepsilon}$ is continuous in $t$ in the sense of convergence $\mu$-a.e.: if $t_{n}\to t$ as $n\to\infty$, then $T_{t_{n}}^{\varepsilon}\to T_{t}^{\varepsilon}$ $\mu$-a.e.

Proof.

We first consider the case where the function $h$ is bounded. We may assume that $h\leq 1$. Let $\varepsilon>0$. Set $\varepsilon_{1}=\varepsilon/5$. Let us take a metrizable compact set $\tilde{K}_{1}\subset X$ such that $\mu(X\setminus\tilde{K}_{1})<\varepsilon_{1}/2$. Since the measure $\mu$ is non-atomic and the compact set $\tilde{K}_{1}$ is metrizable, the measure space $(\tilde{K}_{1},\mu|_{\tilde{K}_{1}})$ is almost homeomorphic to $([0,\mu(\tilde{K}_{1})],\lambda)$, where $\lambda$ is Lebesgue measure (see [5, Theorem 9.6.3]). Let $\varphi\colon[0,\mu(\tilde{K}_{1})]\to\tilde{K}_{1}$ be an almost homeomorphism. Then there exists a compact set $S\subset[0,\mu(\tilde{K}_{1})]$ such that $0<\lambda([0,\mu(\tilde{K}_{1})]\setminus S)<\varepsilon_{1}/2$ and $\varphi|_{S}$ is a homeomorphism. Denote $K_{1}=\varphi(S)$. Then $K_{1}$ is a metrizable compact set and the measure space $(K_{1},\mu|_{K_{1}})$ is homeomorphic to $(S,\lambda)$. Moreover, we have

[TABLE]

Let us take a metrizable compact set $K_{2}\subset Y$ such that $\nu(Y\setminus K_{2})\leq\mu(X\setminus K_{1})$. Let $d_{K_{1}}$ be the metric generating the topology on $K_{1}$.

Let us prove that there exists a continuous (strictly positive) function $\delta\colon T\to(0,+\infty)$ such that for any $x_{1},x_{2}\in K_{1}$, $y\in K_{2}$, $t\in T$ we have $|h(x_{1},y,t)-h(x_{2},y,t)|<\varepsilon_{1}$ if $d_{K_{1}}(x_{1},x_{2})<\delta(t)$. Since $h$ is continuous on $K_{1}\times K_{2}\times T$, it follows that for any $t_{0}\in T$ there exists a real number $\kappa_{t_{0}}>0$ and an open neighbourhood $W_{t_{0}}\subset T$ ($t_{0}\in W_{t_{0}}$) such that $|h(x_{1},y,t)-h(x_{2},y,t)|<\varepsilon_{1}$ for any $x_{1},x_{2}\in K_{1}$ with $d_{K_{1}}(x_{1},x_{2})<\kappa_{t_{0}}$ and for any $y\in K_{2}$, $t\in W_{t_{0}}$. The metric space $T$ posseses a locally finite continuous partition of unity $\{\psi_{\alpha},\alpha\in A\}$ subordinated to the open cover $\{W_{t},t\in T\}$, i.e. a set of continuous functions $\psi_{\alpha}$, $\alpha\in A$, such that $0\leq\psi_{\alpha}\leq 1$ for any $\alpha\in A$, $\operatorname{supp}\psi_{\alpha}\subset W_{\tau(\alpha)}$ for some $\tau(\alpha)\in T$, for every point $t\in T$ there exists a neighbourhood $W$ such that $W\cap\operatorname{supp}\psi_{\alpha}\neq\varnothing$ for at most finite number of indices $\alpha\in A$, and $\sum_{\alpha}\psi_{\alpha}(t)=1$.

Set

[TABLE]

Then the function $\delta(t)$ is continuous, since for any point $t\in T$ there exists a neighbourhood $W$ such that $\delta(t)$ is equal to the sum of a finite number of continuous functions on $W$. Let us show that the function $\delta(t)$ satisfies the required condition. Fix $t_{0}\in T$. Let $\alpha_{1},\dots,\alpha_{N}$ be all indices from the set $A$ such that $\psi_{\alpha_{i}}(t_{0})\neq 0$. Then $t_{0}\in W_{\tau(\alpha_{i})}$ for all $i\in\{1,\dots,N\}$. The equality $\sum_{\alpha}\psi_{\alpha}(t_{0})=1$ implies that $0<\delta(t_{0})\leq\max(\kappa_{\tau(\alpha_{1})},\dots,\kappa_{\tau(\alpha_{N})})$. Therefore, by the definition of the numbers $\kappa_{t}$ we have $|h(x_{1},y,t_{0})-h(x_{2},y,t_{0})|<\varepsilon_{1}$ if $x_{1},x_{2}\in K_{1}$, $d_{K_{1}}(x_{1},x_{2})<\delta(t_{0})$, $y\in K_{2}$.

Let us build a partition

[TABLE]

satisfying the following properties:

for any $j\in\mathbb{N}$ the mapping $t\mapsto I_{S_{j}(t)}$ (where $I_{B}$ denotes the indicator function of a set $B$) is continuous in the sense of convergence $\lambda$-a.e., that is, for any sequence $t_{n}\to t$, $n\to\infty$, we have $I_{S_{j}(t_{n})}\to I_{S_{j}(t)}$ $\lambda$-a.e.,

• 2)

for any $j\in\mathbb{N}$ and for any $t\in T$ we have $|h(\varphi(s_{1}),y,t)-h(\varphi(s_{2}),y,t)|<\varepsilon_{1}$ for all $s_{1},s_{2}\in S_{j}(t)$, $y\in K_{2}$.

Since the mapping $\varphi$ is continuous, as proven above, there exists a continuous function $\tilde{\delta}\colon T\to(0,+\infty)$ such that for any $s_{1},s_{2}\in S$, $y\in K_{2}$, $t\in T$ we have $|h(\varphi(s_{1}),y,t)-h(\varphi(s_{2}),y,t)|<\varepsilon_{1}$ if $|s_{1}-s_{2}|\leq\tilde{\delta}(t)$. Set

[TABLE]

Then $S=\bigsqcup_{j=1}^{\infty}S_{j}(t)$. From the definition of the function $\tilde{\delta}(t)$ it follows that the property 2) is satisfied. Let us prove that the property 1) is fulfilled. Let $t_{n}\to t$ as $n\to\infty$. For any $j\in\mathbb{N}$ let us show that $I_{S_{j}(t_{n})}\to 1$ for all $s\in S\cap((j-1)\tilde{\delta}(t),j\tilde{\delta}(t))$. Fix $s\in S$, $s\in((j-1)\tilde{\delta}(t),j\tilde{\delta}(t))$. Then for all sufficiently large numbers $n$ it holds that $s\in((j-1)\tilde{\delta}(t_{n}),j\tilde{\delta}(t_{n}))$, since $\tilde{\delta}(t_{n})\to\tilde{\delta}(t)$. Therefore, $I_{S_{j}(t_{n})}(s)=1$ for all sufficiently large $n$. Thus for all $s\in S\cap((j-1)\tilde{\delta}(t),j\tilde{\delta}(t))$ and for all $i\in\mathbb{N}$ we have $I_{S_{i}(t_{n})}(s)\to I_{S_{i}(t)}(s)$. Therefore, the property 1) is satisfied.

Set $X_{j}(t)=\varphi(S_{j}(t))$. Then $K_{1}=\bigsqcup_{j=1}^{\infty}X_{j}(t)$. We have $I_{X_{j}(t_{n})}\to I_{X_{j}(t)}$ $\mu$-a.e., if $t_{n}\to t$, $n\to\infty$ (this also implies that $\mu(X_{j}(t_{n})\triangle X_{j}(t))\to 0$ as $n\to\infty$). Furthermore, for any $j\in\mathbb{N}$ and for any $t\in T$ we have $|h(x_{1},y,t)-h(x_{2},y,t)|<\varepsilon_{1}$ for all $x_{1},x_{2}\in X_{j}(t)$, $y\in K_{2}$.

Consider the Kantorovich problem with the cost function $h(x,y,t)$ and measures $\mu|_{K_{1}}$, $\alpha\nu|_{K_{2}}$, where $\alpha=\mu(K_{1})/\nu(K_{2})\leq 1$. By Theorem 2.1 there exist $\varepsilon$-optimal measures $\pi_{t}\in\Pi(\mu|_{K_{1}},\alpha\nu|_{K_{2}})$ for the cost function $h(x,y,t)$ such that $\pi_{t}$ is continuous in $t$ in the weak topology. Let $\nu^{j}_{t}$ be the projection of the measure $I_{X_{j}(t)}\pi_{t}$ on $Y$, $j\in\mathbb{N}$. Let us show that $\nu^{j}_{t}$ is continuous in $t$ in the weak topology. Let $t_{n}\to t$ as $n\to\infty$, we show that the measures $\nu^{j}_{t_{n}}$ converge weakly to $\nu^{j}_{t}$. We have $\|I_{X_{j}(t_{n})}\pi_{t_{n}}-I_{X_{j}(t)}\pi_{t_{n}}\|=\mu(X_{j}(t_{n})\triangle X_{j}(t))\to 0$, where $\|\cdot\|$ is the total variation norm. Therefore, it is sufficient to prove that the measures $I_{X_{j}(t)}\pi_{t_{n}}$ converge weakly to $I_{X_{j}(t)}\pi_{t}$. Let $g\in C_{b}(X\times Y)$, $|g|\leq 1$, we show that

[TABLE]

Fix $\delta>0$. Take a compact set $F_{j}$ and an open set $U_{j}$ such that $F_{j}\subset X_{j}(t)\subset U_{j}$ and $\mu(U_{j}\setminus F_{j})<\delta$. There exists a continuous function $f\colon X\to\mathbb{R}$ such that $f=1$ on $F_{j}$, $f=0$ outside $U_{j}$, $0\leq f\leq 1$. Then

[TABLE]

since $\pi_{t_{n}}$ converge weakly to $\pi_{t}$. Furthermore, we have $|I_{X_{j}(t)}-f(x)|\leq I_{U_{j}\setminus F_{j}}$. Therefore,

[TABLE]

From above we obtain

[TABLE]

Hence $\int g(x,y)I_{X_{j}(t)}\pi_{t_{n}}(dxdy)-\int g(x,y)I_{X_{j}(t)}\pi_{t}(dxdy)\to 0$. Therefore, the measures $\nu^{j}_{t_{n}}$ converge weakly to $\nu^{j}_{t}$, i.e. the mapping $t\mapsto\nu^{j}_{t}$ is continuous in the weak topology.

Since the compact set $K_{2}$ is metrizable, it posseses the strong Skorohod property (see [6]), that is, for any probability measure $\eta$ on $K_{2}$ there exists a mapping $\xi_{\eta}\colon[0,1]\to K_{2}$ such that $\lambda\circ\xi_{\eta}^{-1}=\eta$, where $\lambda$ is Lebesgue measure on $[0,1]$, and if measures $\eta_{n}$ converge weakly to $\eta$, then $\xi_{\eta_{n}}\to\xi_{\eta}$ $\lambda$-a.e.

Since the mapping $t\mapsto\nu^{j}_{t}$ is continuous in the weak topology for any $j\in\mathbb{N}$, by the strong Skorohod property for any $j\in\mathbb{N}$ there exists a mapping $\xi_{t,j}\colon[0,\lambda(S_{j}(t))]\to K_{2}$ such that

[TABLE]

and $\xi_{t,j}$ is continuous in $t$ in the sense of convergence $\lambda$-a.e. Set

[TABLE]

Then the mapping $t\mapsto F^{j}_{t}$ is continuous in $t$ in the topology of pointwise convergence: if $t_{n}\to t$ as $n\to\infty$, then $F^{j}_{t_{n}}(s)\to F^{j}_{t}(s)$ for any $s\in S$. Indeed, $|F^{j}_{t_{n}}(s)-F^{j}_{t}(s)|\leq\lambda(S_{j}(t_{n})\triangle S_{j}(t))\to 0$ as $n\to\infty$. Set

[TABLE]

Then $\mu|_{X_{j}(t)}\circ T_{t}^{-1}=\nu^{j}_{t}$, since $\varphi^{-1}\colon K_{1}\to S$ is a homeomorpism which transfers the measure $\mu|_{X_{j}(t)}$ to the measure $\lambda|_{S_{j}(t)}$ and the mapping $F^{j}_{t}$ transfers the measure $\lambda|_{S_{j}(t)}$ to the measure $\lambda|_{[0,\lambda(S_{j}(t))]}$. Therefore, $\mu|_{K_{1}}\circ T_{t}^{-1}=\alpha\nu|_{K_{2}}$. Since the measure $\mu$ is non-atomic, there exists a mapping $T\colon X\setminus K_{1}\to Y$ such that

[TABLE]

Set $T_{t}(x)=T(x)$ for any $x\in X\setminus K_{1}$. Then $\mu\circ T_{t}^{-1}=\nu$.

Let us show that the mapping $T_{t}$ is continuous in $t$ in the sense of convergence $\mu$-a.e. Let $t_{n}\to t$, $n\to\infty$. Prove that for any $j\in\mathbb{N}$

[TABLE]

For $\mu$-a.e. $x\in X_{j}(t)$ it holds that $x\in X_{j}(t_{n})$ for all sufficiently large $n$, since $I_{X_{j}(t_{n})}\to I_{X_{j}(t)}$ $\mu$-a.e. Therefore, for $\mu$-a.e. $x\in X_{j}(t)$ we have for all sufficiently large $n$

[TABLE]

since $F^{j}_{t_{n}}(\varphi^{-1}(x))\to F^{j}_{t}(\varphi^{-1}(x))$ due to continuity of $F^{j}_{t}$ in $t$ and $\xi_{t_{n},j}\to\xi_{t,j}$ $\lambda$-a.e. Thus $\mu(\{x\in X:T_{t_{n}}(x)\not\to T_{t}(x)\})=0$ and the mapping $T_{t}$ is continuous in $t$ in the sense of convergence $\mu$-a.e.

Let us show that the mapping $T_{t}$ is $\varepsilon$-optimal for every $t\in T$. Fix $t\in T$. For any $j\in\mathbb{N}$ we have (fix some $x_{0}\in X_{j}(t))$

[TABLE]

since $\mu|_{X_{j}(t)}\circ T_{t}^{-1}=\nu_{t}^{j}$ and $|h_{t}(x,y)-h_{t}(x_{0},y)|<\varepsilon_{1}$ for any $x\in X_{j}(t)$, $y\in K_{2}$. Similarly

[TABLE]

Therefore,

[TABLE]

Summing over $j\in\mathbb{N}$, we obtain the inequality

[TABLE]

Moreover, $\int_{X\setminus K_{1}}h_{t}(x,T_{t}x)\mu(dx)\leq\mu(X\setminus K_{1})<\varepsilon_{1}$. Hence

[TABLE]

Let $\sigma\in\Pi(\mu,\nu)$ be an optimal measure in the Kantorovich problem with the cost function $h_{t}(x,y)$ and measures $\mu,\nu$. Let $\mu_{1}$ and $\nu_{1}$ be the projections of the measure $I_{K_{1}\times K_{2}}\sigma$ on $X$ and $Y$ respectively. Set $\tilde{\sigma}=\alpha I_{K_{1}\times K_{2}}\sigma+\zeta$, where $\zeta\in\Pi(\mu|_{K_{1}}-\alpha\mu_{1},\alpha\nu|_{K_{2}}-\alpha\nu_{1})$. Then $\tilde{\sigma}\in\Pi(\mu|_{K_{1}},\alpha\nu|_{K_{2}})$ and hence

[TABLE]

We have $\nu(K_{2})-\nu_{1}(K_{2})=\sigma((X\setminus K_{1})\times K_{2})\leq\mu(X\setminus K_{1})<\varepsilon_{1}$.

Therefore,

[TABLE]

So the mapping $T_{t}$ is $5\varepsilon_{1}$-optimal for any $t\in T$.

Consider now the general case. Let $h(x,y,t)\leq a_{t}(x)+b_{t}(y)$, where the functions $a_{t}\in L^{1}(\mu)$ and $b_{t}\in L^{1}(\nu)$ satisfy (2.2). Let $N\in\mathbb{N}$. As proven above, for the bounded continuous function $\min(h,N)$ there exist $\varepsilon/2$-optimal Monge mappings $T_{t}$ which are continuous in $t$ in the sense of convergence $\mu$-a.e. For any measure $\sigma\in\Pi(\mu,\nu)$ we have

[TABLE]

Take $N\in\mathbb{N}$ such that $\int_{a_{t}\geq N/2}a_{t}d\mu+\int_{b_{t}\geq N/2}b_{t}d\nu<\varepsilon/4$. Then the mappings $T_{t}$ are $\varepsilon$-optimal for the cost function $h$. ∎

3. The Monge problem with marginals depending on the parameter

Assume that the measure $\mu\in\mathcal{P}_{r}(X)$ is fixed and the measures $\nu_{t}\in\mathcal{P}_{r}(Y)$ continuously depend on $t$ in the weak topology. We show that one can select approximate optimal Monge mappings continuously depending on the parameter $t$ in the sense of convergence $\mu$-a.e.

Theorem 3.1**.**

Let $X,Y$ be complete metric spaces and let $\mu$ be a non-atomic Radon probability measure on $X$. Let $T$ be a metric space, the mapping $t\mapsto\nu_{t}$, $T\to\mathcal{P}_{r}(Y)$, is continuous in the weak topology, $h\colon X\times Y\times T\to[0,\infty)$ is a continuous function such that $h(x,y,t)\leq a_{t}(x)+b_{t}(y)$, where $a_{t}\in L^{1}(\mu)$, $b_{t}\in L^{1}(\nu_{t})$ and

[TABLE]

Then for any $\varepsilon>0$ one can select $\varepsilon$-optimal Monge mappings $T_{t}^{\varepsilon}$ for the cost functions $h_{t}$ and measures $\mu$, $\nu_{t}$ (i.e. $\mu\circ(T_{t}^{\varepsilon})^{-1}=\nu_{t}$ for every $t\in T$) such that $T_{t}^{\varepsilon}$ is continuous in $t$ in the sense of convergence $\mu$-a.e.: if $t_{n}\to t$ as $n\to\infty$, then $T_{t_{n}}^{\varepsilon}\to T_{t}^{\varepsilon}$ $\mu$-a.e.

Proof.

The assertion of Theorem 3.1 reduces to the case where $h\leq 1$. Let $\varepsilon>0$. Set $\varepsilon_{1}=\varepsilon/6$. Since the measure $\mu$ is non-atomic, there exists a compact set $K_{1}\subset X$ such that $\mu(X\setminus K_{1})<\varepsilon_{1}$ and $(K_{1},\mu|_{K_{1}})$ is homeomorphic to $(S,\lambda)$, where $S\subset[0,1]$ is a compact set and $\lambda$ is Lebesgue measure. Let $\varphi\colon S\to K_{1}$ be a homeomorphism, $\lambda|_{S}\circ\varphi^{-1}=\mu|_{K_{1}}$. Let $d_{X}$ and $d_{Y}$ be the metrics of $X$ and $Y$ respectively.

Let us prove that there exists a continuous (strictly positive) function $\delta\colon T\to(0,+\infty)$ and a collection of closed sets $Y(t)\subset Y$, $t\in T$, such that for any $t\in T$ we have $\nu_{t}(Y\setminus Y(t))<\varepsilon_{1}$ and $|h(x_{1},y,t)-h(x_{2},y,t)|<\varepsilon_{1}$ for all $x_{1},x_{2}\in K_{1}$ with $d_{X}(x_{1},x_{2})<\delta(t)$ and for all $y\in Y(t)$.

For any $t\in T$ take a compact set $K_{2}(t)\subset Y$ such that $\nu_{t}(Y\setminus K_{2}(t))<\varepsilon_{1}$. Since $h$ is continuous on $K_{1}\times Y\times T$, it follows that for any $t_{0}\in T$ there exist real numbers $\kappa(t_{0})>0$, $r(t_{0})>0$ and an open neighbourhood $\tilde{W}_{t_{0}}\subset T$ ($t_{0}\in\tilde{W}_{t_{0}}$) such that $|h(x_{1},y,t)-h(x_{2},y,t)|<\varepsilon_{1}$ for any $x_{1},x_{2}\in K_{1}$ with $d_{X}(x_{1},x_{2})<\kappa(t_{0})$ and for any $y\in K_{2}(t_{0})^{r(t_{0})}$ (where $B^{r}=\{y\in Y:d_{Y}(y,B)\leq r\}$ is a closed $r$-neighbourhood of a set $B$ in the metric space $Y$), $t\in\tilde{W}_{t_{0}}$. Since the mapping $t\mapsto\nu_{t}$ is continuous in the weak topology and $\nu_{t_{0}}(Y\setminus K_{2}(t_{0}))<\varepsilon_{1}$, there exists an oper neighbourhood $W^{\prime}_{t_{0}}\subset T$ ($t_{0}\in W^{\prime}_{t_{0}}$) such that $\nu_{t}(Y\setminus K_{2}(t_{0})^{r(t_{0})})<\varepsilon_{1}$ for any $t\in W^{\prime}_{t_{0}}$. Set $W_{t_{0}}=\tilde{W}_{t_{0}}\cap W^{\prime}_{t_{0}}$.

The metric space $T$ posseses a locally finite continuous partition of unity $\{\psi_{\alpha},\alpha\in A\}$ subordinated to the open cover $\{W_{t},t\in T\}$, i.e. a set of continuous functions $\psi_{\alpha}$, $\alpha\in A$, such that $0\leq\psi_{\alpha}\leq 1$ for any $\alpha\in A$, $\operatorname{supp}\psi_{\alpha}\subset W_{\tau(\alpha)}$ for some $\tau(\alpha)\in T$, for every point $t\in T$ there exists a neighbourhood $W$ such that $W\cap\operatorname{supp}\psi_{\alpha}\neq\varnothing$ for at most finite number of indices $\alpha\in A$, and $\sum_{\alpha}\psi_{\alpha}(t)=1$.

Set

[TABLE]

Then the function $\delta(t)$ is continuous, since for any point $t\in T$ there exists a neighbourhood $W$ such that $\delta(t)$ is equal to the sum of a finite number of continuous functions on $W$. For any $t\in T$ choose an index $\alpha(t)$ from the finite set $\{\alpha\in A:\psi_{\alpha}(t)\neq 0\}$ for which the value $\kappa(\tau(\alpha))$ is maximal. Set

[TABLE]

Let us show that the function $\delta(t)$ and the sets $Y(t)$, $t\in T$, satisfy the required condition. Fix $t_{0}\in T$. Let $\alpha_{1},\dots,\alpha_{N}$ be all indices from the set $A$ such that $\psi_{\alpha_{i}}(t_{0})\neq 0$. Then $t_{0}\in W_{\tau(\alpha_{i})}$ for all $i\in\{1,\dots,N\}$. Since $\sum_{\alpha}\psi_{\alpha}(t_{0})=1$, we have $\delta(t_{0})\leq\max(\kappa(\tau(\alpha_{1})),\dots,\kappa(\tau(\alpha_{N})))=\kappa(\tau(\alpha(t_{0})))$. Therefore, by the definition of the numbers $\kappa(t)$ we obtain that $|h(x_{1},y,t_{0})-h(x_{2},y,t_{0})|<\varepsilon_{1}$ if $x_{1},x_{2}\in K_{1}$, $d_{X}(x_{1},x_{2})<\delta(t_{0})$, $y\in Y(t_{0})$. Moreover, $\nu_{t_{0}}(Y\setminus Y(t_{0}))<\varepsilon_{1}$, because $t_{0}\in W_{\tau(\alpha(t_{0}))}$.

Since the mapping $\varphi$ is continuous, as proven above, there exists a continuous function $\tilde{\delta}\colon T\to(0,+\infty)$ and a collection of closed sets $Y(t)\subset Y$, $t\in T$, such that for any $t\in T$ we have $\nu_{t}(Y\setminus Y(t))<\varepsilon_{1}$ and $|h(\varphi(s_{1}),y,t)-h(\varphi(s_{2}),y,t)|<\varepsilon_{1}$ for all $s_{1},s_{2}\in S$ with $|s_{1}-s_{2}|\leq\tilde{\delta}(t)$ and for all $y\in Y(t)$.

As described in the proof of Theorem 2.2, we can construct a partition $S\leavevmode\nobreak\ =\leavevmode\nobreak\ \bigsqcup_{j=1}^{\infty}S_{j}(t)$ satisfying the following properties:

for any $j\in\mathbb{N}$ the mapping $t\mapsto I_{S_{j}(t)}$ is continuous in the sense of convergence $\lambda$-a.e., that is, for any sequence $t_{n}\to t$, $n\to\infty$, we have $I_{S_{j}(t_{n})}\to I_{S_{j}(t)}$ $\lambda$-a.e.,

• 2)

for any $j\in\mathbb{N}$ and for any $t\in T$ we have $|h(\varphi(s_{1}),y,t)-h(\varphi(s_{2}),y,t)|<\varepsilon_{1}$ for all $s_{1},s_{2}\in S_{j}(t)$, $y\in Y(t)$.

Set $X_{j}(t)=\varphi(S_{j}(t))$. Then $K_{1}=\bigsqcup_{j=1}^{\infty}X_{j}(t)$. We have $I_{X_{j}(t_{n})}\to I_{X_{j}(t)}$ $\mu$-a.e., if $t_{n}\to t$, $n\to\infty$ (this also implies that $\mu(X_{j}(t_{n})\triangle X_{j}(t))\to 0$ as $n\to\infty$). Furthermore, for any $j\in\mathbb{N}$ and for any $t\in T$ we have $|h(x_{1},y,t)-h(x_{2},y,t)|<\varepsilon_{1}$ for all $x_{1},x_{2}\in X_{j}(t)$, $y\in Y(t)$. Set $X_{0}(t)=X\setminus K_{1}$.

By Theorem 2.1 there exist $\varepsilon_{1}$-optimal measures $\pi_{t}\in\Pi(\mu,\nu_{t})$ for the cost function $h(x,y,t)$ such that $\pi_{t}$ is continuous in $t$ in the weak topology. Let $\nu^{j}_{t}$ be the projection of the measure $I_{X_{j}(t)}\pi_{t}$ on $Y$, $j\in\mathbb{N}\cup\{0\}$. Then $\nu^{j}_{t}$ is continuous in $t$ in the weak topology. Indeed, if $t_{n}\to t$ as $n\to\infty$, then the measures $\nu^{j}_{t_{n}}$ converge weakly to $\nu^{j}_{t}$, since the measures $\pi_{t_{n}}$ converge weakly to $\pi_{t}$ and $\mu(X_{j}(t_{n})\triangle X_{j}(t))\to 0$.

The complete metric space $Y$ posseses the strong Skorohod property for Radon measures (see [6]), that is, for any Radon probability measure $\eta$ on $Y$ there exists a mapping $\xi_{\eta}\colon[0,1]\to Y$ such that $\lambda\circ\xi_{\eta}^{-1}=\eta$, where $\lambda$ is Lebesgue measure on $[0,1]$, and if measures $\eta_{n}$ converge weakly to $\eta$, then $\xi_{\eta_{n}}\to\xi_{\eta}$ $\lambda$-a.e.

Since the mapping $t\mapsto\nu^{j}_{t}$ is continuous in the weak topology for any $j\in\mathbb{N}\cup\{0\}$, by the strong Skorohod property for any $j\in\mathbb{N}\cup\{0\}$ there exists a mapping $\xi_{t,j}\colon[0,\mu(X_{j}(t))]\to Y$ (where $\mu(X_{j}(t))=\lambda(S_{j}(t))$ for any $j\in\mathbb{N}$ and $\mu(X_{0}(t))=\mu(X\setminus K_{1})$) such that

[TABLE]

and $\xi_{t,j}$ is continuous in $t$ in the sense of convergence $\lambda$-a.e. Let

[TABLE]

The mapping $t\mapsto F^{j}_{t}$ is continuous in $t$ in the topology of pointwise convergence: if $t_{n}\to t$ as $n\to\infty$, then $F^{j}_{t_{n}}(s)\to F^{j}_{t}(s)$ for any $s\in S$. Indeed, $|F^{j}_{t_{n}}(s)-F^{j}_{t}(s)|\leq\lambda(S_{j}(t_{n})\triangle S_{j}(t))\to 0$ as $n\to\infty$. Set

[TABLE]

Then $\mu|_{X_{j}(t)}\circ T_{t}^{-1}=\nu^{j}_{t}$, since $\varphi^{-1}\colon K_{1}\to S$ is a homeomorphism which transfers the measure $\mu|_{X_{j}(t)}$ to the measure $\lambda|_{S_{j}(t)}$ and the mapping $F^{j}_{t}$ transfers $\lambda|_{S_{j}(t)}$ to the measure $\lambda|_{[0,\lambda(S_{j}(t))]}$. Since the measure $\mu$ is non-atomic, there exists a mapping $F\colon X\setminus K_{1}\to[0,\mu(X\setminus K_{1})]$ such that

[TABLE]

Set $T_{t}(x)=\xi_{t,0}(F(x))$ for any $x\in X\setminus K_{1}$. Then $\mu|_{X\setminus K_{1}}\circ T_{t}^{-1}=\nu_{t}^{0}$. Therefore, $\mu\circ T_{t}^{-1}=\nu_{t}$ for any $t\in T$.

Let us show that the mapping $T_{t}$ is continuous in $t$ in the sense of convergence $\mu$-a.e. Let $t_{n}\to t$, $n\to\infty$. Prove that for any $j\in\mathbb{N}$

[TABLE]

Indeed, for $\mu$-a.e. $x\in X_{j}(t)$ it holds that $x\in X_{j}(t_{n})$ for all sufficiently large $n$, since $I_{X_{j}(t_{n})}\to I_{X_{j}(t)}$ $\mu$-a.e. Therefore, for $\mu$-a.e. $x\in X_{j}(t)$ for all sufficiently large $n$ we have

[TABLE]

since $F^{j}_{t_{n}}(\varphi^{-1}(x))\to F^{j}_{t}(\varphi^{-1}(x))$ due to the continuity of $F^{j}_{t}$ in $t$ and $\xi_{t_{n},j}\to\xi_{t,j}$ $\lambda$-a.e. Moreover,

[TABLE]

Therofore, $\mu(\{x\in X:T_{t_{n}}(x)\not\to T_{t}(x)\})=0$ and the mapping $T_{t}$ is continuous in $t$ in the sense of convergence $\mu$-a.e.

Let us prove that the mapping $T_{t}$ is $\varepsilon$-optimal for any $t\in T$. Fix $t\in T$. For any $j\in\mathbb{N}$ we have (fix some $x_{0}\in X_{j}(t))$

[TABLE]

since $\mu|_{X_{j}(t)}\circ T_{t}^{-1}=\nu^{j}_{t}$ and $|h_{t}(x,y)-h_{t}(x_{0},y)|<\varepsilon_{1}$ for any $x\in X_{j}(t)$, $y\in Y(t)$. Similarly

[TABLE]

Therefore,

[TABLE]

Summing over $j\in\mathbb{N}$, we obtain the inequality

[TABLE]

Furthermore,

[TABLE]

Therefore,

[TABLE]

Thus the mapping $T_{t}$ is $6\varepsilon_{1}$-optimal for every $t\in T$. ∎

Corollary 3.2**.**

The statement of Theorem 3.1 holds true if we replace the condition that $X$ is a complete metric space by the condition that $X$ is a completely regular topological space and the measure $\mu$ is concentrated on a countable union of metrizable compact sets (i.e. we may assume that $X$ is a Souslin space).

Proof.

Following the proof of Theorem 3.1 we construct the sets $Y(t)$ and partitions $K_{1}=\bigsqcup_{j=1}^{\infty}X_{j}(t)$, $t\in T$. According to Theorem 2.1, consider $\varepsilon_{1}$-optimal measures $\pi_{t}\leavevmode\nobreak\ \in\leavevmode\nobreak\ \Pi(\mu|_{K_{1}},\mu(K_{1})\nu)$ in the Kantorovich problem for the measures $\mu|_{K_{1}}$ and $\mu(K_{1})\nu$ with the cost function $h(x,y,t)$ such that $\pi_{t}$ is continuous in $t$ in the weak topology. Set $\nu^{j}_{t}=I_{X_{j}(t)}\pi_{t}$ for any $j\in\mathbb{N}$. Then $\nu^{j}_{t}$ is continuous in $t$ in the weak topology. Define the mapping $T_{t}$ on $K_{1}$ in the same way as in the proof of Theorem 3.1, then we have $\mu|_{K_{1}}\circ T_{t}^{-1}=\mu(K_{1})\nu_{t}$. Take a mapping $F\colon X\setminus K_{1}\to[0,\mu(X\setminus K_{1})]$ such that

[TABLE]

Set $T_{t}(x)=\xi_{t}(F(x))$ for any $x\in X\setminus K_{1}$, where $\xi_{t}\colon[0,\mu(X\setminus K_{1})]\to Y$,

[TABLE]

and $\xi_{t}$ is continuous in $t$ in the sense of convergence $\lambda$-a.e. Then $\mu\circ T_{t}^{-1}=\nu_{t}$, $T_{t}$ is continuous in $t$ in the sense of convergence $\mu$-a.e. and $T_{t}$ is $\varepsilon$-optimal for every $t\in T$. ∎

Consider now the most general case where the measures $\mu_{t}\in\mathcal{P}_{r}(X)$ and $\nu_{t}\in\mathcal{P}_{r}(Y)$ continuously depend on $t$. Assuming that the measures $\mu_{t}$ are continuous in $t$ in the total variation norm we prove the existence of approximate optimal Monge mappings continuously depending on the parameter $t$ in the sense of convergence $\mu_{t}$-a.e.

Theorem 3.3**.**

Let $X$ be a complete separable metric space and let $Y$ be a complete metric space. Let $T$ be a metric space, the mapping $t\mapsto\nu_{t}$, $T\to\mathcal{P}_{r}(Y)$, is continuous in the weak topology, the mapping $t\mapsto\mu_{t}$, $T\to\mathcal{P}_{r}(X)$, is continuous in the total variation norm, and the measures $\mu_{t}$ are non-atomic for all $t\in T$. Let $h\colon X\times Y\times T\to[0,\infty)$ be a continuous function such that $h(x,y,t)\leq a_{t}(x)+b_{t}(y)$, where $a_{t}\in L^{1}(\mu_{t})$, $b_{t}\in L^{1}(\nu_{t})$ and

[TABLE]

Then for any $\varepsilon>0$ one can select $\varepsilon$-optimal Monge mappings $T_{t}^{\varepsilon}$ for the cost functions $h_{t}$ and measures $\mu_{t}$, $\nu_{t}$ (i.e. $\mu_{t}\circ(T_{t}^{\varepsilon})^{-1}=\nu_{t}$ for every $t\in T$) such that $T_{t}^{\varepsilon}$ is continuous in $t$ in the sense of convergence $\mu_{t}$-a.e.: if $t_{n}\to t$ as $n\to\infty$, then $T_{t_{n}}^{\varepsilon}\to T_{t}^{\varepsilon}$ $\mu_{t}$-a.e.

Proof.

The assertion of Theorem 3.3 reduces to the case where $h\leq 1$. Let $\varepsilon>0$. Set $\varepsilon_{1}=\varepsilon/7$. Since every complete separable metric space is homeomorphic to a $G_{\delta}$-set in $[0,1]^{\infty}$ (see [15]), we may assume that $X\subset[0,1]^{\infty}$. The compact metrizable space $[0,1]^{\infty}$ is a continuous image of the Cantor set $C$, i.e. there exists a surjective continuous mapping $f\colon C\to[0,1]^{\infty}$. By measurable selection theorem (see [5]) there exists a Borel measurable mapping $g\colon[0,1]^{\infty}\to C$ such that $f(g(x))=x$ for all $x\in[0,1]^{\infty}$. Set $\gamma_{t}=\mu_{t}\circ g^{-1}$, $t\in T$. Then $\mu_{t}=\gamma_{t}\circ f^{-1}$ for every $t\in T$ and the measures $\gamma_{t}$ are non-atomic. Moreover, the mapping $t\mapsto\gamma_{t}$ is continuous in the total variation norm, since $\|\gamma_{t}-\gamma_{\tau}\|=\|(\mu_{t}-\mu_{\tau})\circ g^{-1}\|\leq\|\mu_{t}-\mu_{\tau}\|$ for any $t,\tau\in T$. Set $S=g(X)$. Then $S$ is a Borel subset of $C$. Let $d_{X}$ and $d_{Y}$ be the metrics on $X$ and $Y$ respectively.

Let us prove that there exists a continuous (strictly positive) function $\delta\colon T\to(0,+\infty)$ and a collection of compact sets $X(t)\subset X$ and closed sets $Y(t)\subset Y$, $t\in T$, such that for any $t\in T$ we have $\mu_{t}(X\setminus X(t))<\varepsilon_{1}$, $\nu_{t}(Y\setminus Y(t))<\varepsilon_{1}$ and $|h(x_{1},y,t)-h(x_{2},y,t)|<\varepsilon_{1}$ for any $x_{1},x_{2}\in X(t)$ with $d_{X}(x_{1},x_{2})<\delta(t)$ and for any $y\in Y(t)$.

For every $t\in T$ take compact sets $K_{1}(t)\subset X$ and $K_{2}(t)\subset Y$ such that $\mu_{t}(X\setminus K_{1}(t))<\varepsilon_{1}$ and $\nu_{t}(Y\setminus K_{2}(t))<\varepsilon_{1}$. Since $h$ is continuous on $X\times Y\times T$, for any $t_{0}\in T$ there exist real numbers $\kappa(t_{0})>0$, $r(t_{0})>0$ and an open neighbourhood $\tilde{W}_{t_{0}}\subset T$ ($t_{0}\in\tilde{W}_{t_{0}}$) such that $|h(x_{1},y,t)-h(x_{2},y,t)|<\varepsilon_{1}$ for any $x_{1},x_{2}\in K_{1}(t_{0})$ with $d_{X}(x_{1},x_{2})<\kappa(t_{0})$ and for any $y\in K_{2}(t_{0})^{r(t_{0})}$ (where $B^{r}=\{y\in Y:d_{Y}(y,B)\leq r\}$ is a closed $r$-neighbourhood of a set $B$ in the metric space $Y$), $t\in\tilde{W}_{t_{0}}$. Since the mapping $t\mapsto\nu_{t}$ is continuous in the weak topology and $\nu_{t_{0}}(Y\setminus K_{2}(t_{0}))<\varepsilon_{1}$, there exists an open neighbourhood $W^{\prime}_{t_{0}}\subset T$ ($t_{0}\in W^{\prime}_{t_{0}}$) such that $\nu_{t}(Y\setminus K_{2}(t_{0})^{r(t_{0})})<\varepsilon_{1}$ for any $t\in W^{\prime}_{t_{0}}$. Since the mapping $t\mapsto\mu_{t}$ is continuous in the total variation norm, there exists an open neighbourhood $W^{\prime\prime}_{t_{0}}\subset T$ ($t_{0}\in W^{\prime\prime}_{t_{0}}$) such that $\mu_{t}(X\setminus K_{1}(t_{0}))<\varepsilon_{1}$ for any $t\in W^{\prime\prime}_{t_{0}}$. Set $W_{t_{0}}=\tilde{W}_{t_{0}}\cap W^{\prime}_{t_{0}}\cap W^{\prime\prime}_{t_{0}}$.

The metric space $T$ posseses a locally finite continuous partition of unity $\{\psi_{\alpha},\alpha\in A\}$ subordinated to the open cover $\{W_{t},t\in T\}$, i.e. a set of continuous functions $\psi_{\alpha}$, $\alpha\in A$, such that $0\leq\psi_{\alpha}\leq 1$ for any $\alpha\in A$, $\operatorname{supp}\psi_{\alpha}\subset W_{\tau(\alpha)}$ for some $\tau(\alpha)\in T$, for every point $t\in T$ there exists a neighbourhood $W$ such that $W\cap\operatorname{supp}\psi_{\alpha}\neq\varnothing$ for at most finite number of indices $\alpha\in A$, and $\sum_{\alpha}\psi_{\alpha}(t)=1$.

Set

[TABLE]

Then the function $\delta(t)$ is continuous, since for any point $t\in T$ there exists a neighbourhood $W$ such that $\delta(t)$ is equal to the sum of a finite number of continuous functions on $W$. For any $t\in T$ choose an index $\alpha(t)$ from the finite set $\{\alpha\in A:\psi_{\alpha}(t)\neq 0\}$ for which the value $\kappa(\tau(\alpha))$ is maximal. Set

[TABLE]

Let us show that the function $\delta(t)$ and the sets $X(t)$, $Y(t)$, $t\in T$, satisfy the required condition. Fix $t_{0}\in T$. Let $\alpha_{1},\dots,\alpha_{N}$ be all indices from the set $A$ such that $\psi_{\alpha_{i}}(t_{0})\neq 0$. Then $t_{0}\in W_{\tau(\alpha_{i})}$ for all $i\in\{1,\dots,N\}$. Since $\sum_{\alpha}\psi_{\alpha}(t_{0})=1$, we have $\delta(t_{0})\leq\max(\kappa(\tau(\alpha_{1})),\dots,\kappa(\tau(\alpha_{N})))=\kappa(\tau(\alpha(t_{0})))$. Therefore, by the definition of the numbers $\kappa(t)$ we obtain that $|h(x_{1},y,t_{0})-h(x_{2},y,t_{0})|<\varepsilon_{1}$ if $x_{1},x_{2}\in X(t_{0})$, $d_{X}(x_{1},x_{2})<\delta(t_{0})$, $y\in Y(t_{0})$. Moreover, $\mu_{t_{0}}(X\setminus X(t_{0}))<\varepsilon_{1}$ and $\nu_{t_{0}}(Y\setminus Y(t_{0}))<\varepsilon_{1}$, because $t_{0}\in W_{\tau(\alpha(t_{0}))}$.

Since the mapping $f$ is continuous, the function $h(f(s),y,t)$ is continuous on $S\times Y\times T$. As proven above, there exists a continuous function $\tilde{\delta}\colon T\to(0,+\infty)$ and a collection of sets $S(t)\subset S$, $Y(t)\subset Y$, $t\in T$, such that for any $t\in T$ we have $\gamma_{t}(S\setminus S(t))<\varepsilon_{1}$, $\nu_{t}(Y\setminus Y(t))<\varepsilon_{1}$ and $|h(f(s_{1}),y,t)-h(f(s_{2}),y,t)|<\varepsilon_{1}$ for all $s_{1},s_{2}\in S(t)$ with $|s_{1}-s_{2}|\leq\tilde{\delta}(t)$ and for all $y\in Y(t)$.

As described in the proof of Theorem 2.2, we can construct a partition $S\leavevmode\nobreak\ =\leavevmode\nobreak\ \bigsqcup_{j=1}^{\infty}S_{j}(t)$ satisfying the following properties:

for any $j\in\mathbb{N}$ the mapping $t\mapsto I_{S_{j}(t)}$ is continuous in the sense of convergence $\gamma_{t}$-a.e., that is, for any sequence $t_{n}\to t$, $n\to\infty$, we have $I_{S_{j}(t_{n})}\to I_{S_{j}(t)}$ $\gamma_{t}$-a.e.,

• 2)

for any $j\in\mathbb{N}$ and for any $t\in T$ we have $|h(f(s_{1}),y,t)-h(f(s_{2}),y,t)|<\varepsilon_{1}$ for all $s_{1},s_{2}\in S(t)\cap S_{j}(t)$, $y\in Y(t)$.

Set $X(t)=f(S(t))$ and $X_{j}(t)=f(S_{j}(t))$, $j\in\mathbb{N}$. Then $X=\bigsqcup_{j=1}^{\infty}X_{j}(t)$. We have $I_{X_{j}(t_{n})}\to I_{X_{j}(t)}$ $\mu_{t}$-a.e.., if $t_{n}\to t$, $n\to\infty$ (this also implies that $\mu_{t}(X_{j}(t_{n})\triangle X_{j}(t))\to 0$ as $n\to\infty$). Furthermore, for any $j\in\mathbb{N}$ and for any $t\in T$ we have $|h(x_{1},y,t)-h(x_{2},y,t)|<\varepsilon_{1}$ for all $x_{1},x_{2}\in X(t)\cap X_{j}(t)$, $y\in Y(t)$.

By Theorem 2.1 there exist $\varepsilon_{1}$-optimal measures $\pi_{t}\in\Pi(\mu_{t},\nu_{t})$ for the cost function $h(x,y,t)$ such that $\pi_{t}$ is continuous in $t$ in the weak topology. Let $\nu^{j}_{t}$ be the projection of the measure $I_{X_{j}(t)}\pi_{t}$ on $Y$, $j\in\mathbb{N}$. Let us show that $\nu^{j}_{t}$ is continuous in $t$ in the weak topology. Let $t_{n}\to t$ as $n\to\infty$, we show that the measures $\nu^{j}_{t_{n}}$ converge weakly to $\nu^{j}_{t}$. We have

[TABLE]

since the mapping $t\mapsto\mu_{t}$ is continuous in the total variation norm. Let us prove that the measures $I_{X_{j}(t)}\pi_{t_{n}}$ converge weakly to $I_{X_{j}(t)}\pi_{t}$. Let $\zeta\in C_{b}(X\times Y)$, $|\zeta|\leq 1$, we show that

[TABLE]

Fix $\delta>0$. Take a compact set $F_{j}$ and an open set $U_{j}$ such that $F_{j}\subset X_{j}(t)\subset U_{j}$ and $\mu_{t}(U_{j}\setminus F_{j})<\delta$. There exist a continuous function $\chi\colon X\to\mathbb{R}$ such that $\chi=1$ on $F_{j}$, $\chi=0$ outside $U_{j}$, $0\leq\chi\leq 1$. Then

[TABLE]

since the measures $\pi_{t_{n}}$ converge weakly to $\pi_{t}$. Furthermore,

[TABLE]

since $|I_{X_{j}(t)}-\chi|\leq I_{U_{j}\setminus F_{j}}$ and $|\zeta|\leq 1$. Therefore,

[TABLE]

Hence we obtain that $\int_{X\times Y}\zeta(x,y)I_{X_{j}(t)}\pi_{t_{n}}(dxdy)-\int_{X\times Y}\zeta(x,y)I_{X_{j}(t)}\pi_{t}(dxdy)\to 0$. Therefore, the measures $\nu^{j}_{t_{n}}$ converge weakly to $\nu^{j}_{t}$, i.e. the mapping $t\mapsto\nu^{j}_{t}$ is continuous in $t$ in the weak topology.

The complete metric space $Y$ posseses the strong Skorohod property for Radon measures, that is, for any Radon probability measure $\eta$ on $Y$ there exists a mapping $\xi_{\eta}\colon[0,1]\to Y$ such that $\lambda\circ\xi_{\eta}^{-1}=\eta$, where $\lambda$ is Lebesgue measure on $[0,1]$, and if measures $\eta_{n}$ converge weakly to $\eta$, then $\xi_{\eta_{n}}\to\xi_{\eta}$ $\lambda$-a.e.

Since the mapping $t\mapsto\nu^{j}_{t}$ is continuous in the weak topology for any $j\in\mathbb{N}$, by the strong Skorohod property for any $j\in\mathbb{N}$ there exists a mapping $\xi_{t,j}\colon[0,\mu_{t}(X_{j}(t))]\to Y$ (where $\mu_{t}(X_{j}(t))=\gamma_{t}(S_{j}(t))$ for any $j\in\mathbb{N}$) such that

[TABLE]

and $\xi_{t,j}$ is continuous in $t$ in the sense of convergence $\lambda$-a.e. Let

[TABLE]

The mapping $t\mapsto F^{j}_{t}$ is continuous in $t$ in the topology of pointwise convergence: if $t_{n}\to t$ as $n\to\infty$, then $F^{j}_{t_{n}}(s)\to F^{j}_{t}(s)$ for any $s\in S$. Indeed,

[TABLE]

Set

[TABLE]

Then $\mu_{t}|_{X_{j}(t)}\circ T_{t}^{-1}=\nu^{j}_{t}$, since the mapping $g$ transfers the measure $\mu_{t}|_{X_{j}(t)}$ to the measure $\gamma_{t}|_{S_{j}(t)}$ and the mapping $F^{j}_{t}$ transfers the measure $\gamma_{t}|_{S_{j}(t)}$ to the measure $\lambda|_{[0,\mu_{t}(X_{j}(t))]}$. Therefore, $\mu_{t}\circ T_{t}^{-1}=\nu_{t}$ for any $t\in T$.

Let us show that the mapping $T_{t}$ is continuous in $t$ in the sense of convergence $\mu_{t}$-a.e. Let $t_{n}\to t$, $n\to\infty$. Prove that for any $j\in\mathbb{N}$

[TABLE]

Indeed, for $\mu_{t}$-a.e. $x\in X_{j}(t)$ it holds that $x\in X_{j}(t_{n})$ for all sufficiently large $n$, since $I_{X_{j}(t_{n})}\to I_{X_{j}(t)}$ $\mu_{t}$-a.e. Therefore, for $\mu_{t}$-a.e. $x\in X_{j}(t)$ for all sufficiently large $n$ we have

[TABLE]

since $F^{j}_{t_{n}}(g(x))\to F^{j}_{t}(g(x))$ due to the continuity of $F^{j}_{t}$ in $t$ and $\xi_{t_{n},j}\to\xi_{t,j}$ $\lambda$-a.e. Therofore, $\mu_{t}(\{x\in X:T_{t_{n}}(x)\not\to T_{t}(x)\})=0$ and the mapping $T_{t}$ is continuous in $t$ in the sense of convergence $\mu_{t}$-a.e.

Let us prove that the mapping $T_{t}$ is $\varepsilon$-optimal for any $t\in T$. Fix $t\in T$. For any $j\in\mathbb{N}$ we have (fix some $x_{0}\in X_{j}(t)\cap X(t))$

[TABLE]

since $\mu_{t}|_{X_{j}(t)}\circ T_{t}^{-1}=\nu^{j}_{t}$ and $|h_{t}(x,y)-h_{t}(x_{0},y)|<\varepsilon_{1}$ for any $x\in X_{j}(t)\cap X(t)$, $y\in Y(t)$. Similarly

[TABLE]

Therefore,

[TABLE]

Summing over $j\in\mathbb{N}$, we obtain that

[TABLE]

Therefore, the mapping $T_{t}$ is $7\varepsilon_{1}$-optimal for every $t\in T$. ∎

Corollary 3.4**.**

The statement of Theorem 3.3 holds true in the case where $X$ is a Souslin space.

Proof.

The Souslin space $X$ is an image of a complete separable metric space $\tilde{X}$ under a continuous surjective mapping $f\colon\tilde{X}\to X$. By measurable selection theorem (see [5]) there exists a mapping $g\colon X\to\tilde{X}$ such that $g$ is measurable with respect to the $\sigma$-algebra generated by Souslin sets and $f(g(x))=x$ for all $x\in X$. Set $\gamma_{t}=\mu_{t}\circ g^{-1}$ for any $t\in T$. Then $\mu_{t}=\gamma_{t}\circ f^{-1}$ and the measures $\gamma_{t}$ are non-atomic. The mapping $t\mapsto\gamma_{t}$ is continuous in the total variation norm, since $\|\gamma_{t}-\gamma_{\tau}\|=\|\mu_{t}-\mu_{\tau}\|$ for any $t,\tau\in T$. The function $h(f(\tilde{x}),y,t)$ is continuous on $\tilde{X}\times Y\times T$. Consider the Kantorovich problem with the cost function $h(f(\tilde{x}),y,t)$ and measures $\gamma_{t}$, $\nu_{t}$, $t\in T$. By Theorem 3.3 there exist $\varepsilon$-optimal mappings $\tilde{T}_{t}\colon\tilde{X}\to Y$ such that $\tilde{T}_{t}$ is continuous in $t$ in the sense of convergence $\gamma_{t}$-a.e. Set $T_{t}(x)=\tilde{T}_{t}(g(x))$. Then $\mu_{t}\circ T_{t}^{-1}=\gamma_{t}\circ\tilde{T}_{t}^{-1}=\nu_{t}$ for any $t\in T$. The mapping $t\mapsto T_{t}$ is continuous in $t$ in the sense of convergence $\mu_{t}$-a.e. Indeed, if $t_{n}\to t$, $n\to\infty$, then

[TABLE]

Let us show that the mapping $T_{t}$ is $\varepsilon$-optimal for any $t\in T$. We have

[TABLE]

Let $\sigma\in\Pi(\mu_{t},\nu_{t})$ be an optimal plan in the Kantorovich problem with the cost function $h(x,y,t)$ and measures $\mu_{t},\nu_{t}$. Let $\tilde{\sigma}$ be the image of the measure $\sigma$ under the mapping $(x,y)\mapsto(g(x),y)$. Then $\tilde{\sigma}\in\Pi(\gamma_{t},\nu_{t})$ and

[TABLE]

Therefore, the minimum in the Kantorovich problem with the cost function $h(f(\tilde{x}),y,t)$ and measures $\gamma_{t},\nu_{t}$ equals the minimum in the Kantorovich problem with the cost function $h(x,y,t)$ and measures $\mu_{t},\nu_{t}$. Therefore, the mapping $T_{t}$ is $\varepsilon$-optimal. ∎