$\newcommand{\diff}{ \, \mathrm d}$


Let


* $X,Y$ be Polish spaces,
* $\mathcal C\_b(X)$ the space of all real-valued bounded continuous functions on $X$,
* $\mathcal P(X)$ the space of Borel probability measures on $X$,
* $\mu \in \mathcal P(X)$ and $\nu \in \mathcal P(Y)$.
* $L\_1 (\mu)$ the space of all $\mu$-integrable functions $\varphi:X \to \mathbb R \cup \{-\infty\}$,
* $\Pi(\mu, \nu)$ a subset of $\mathcal P(X \times Y)$ that contains all measures whose marginal on $X$ is $\mu$ and that on $Y$ is $\nu$, and
* $c:X \times Y \to [0, +\infty]$ measurable.


Let $\Phi\_c$ (resp. $\Phi'\_c$) be the collection of all $(\varphi, \psi) \in \mathcal C\_b(X) \times \mathcal C\_b(Y)$ (resp. $(\varphi, \psi) \in L\_1 (\mu) \times L\_1 (\nu)$) such that $\varphi (x)+\psi(y) \le c(x, y)$ for all $(x,y) \in X \times Y$. Let
$$
\begin{align}
\mathbb J (\varphi, \psi) &:= \int \varphi \diff \mu + \int \psi \diff \nu &&\forall (\varphi, \psi) \in L\_1(\mu) \times L\_1(\nu),\\
\mathbb K (\gamma) &:= \int c \diff \gamma &&\forall \gamma \in \Pi(\mu, \nu).
\end{align}
$$


The Kantorovich and its dual problems are
$$
\begin{align}
(\mathrm{KP}) &: \quad \inf \left \{ \mathbb K (\gamma) : \gamma \in \Pi(\mu, \nu) \right \}, \\
(\mathrm{DP}) &: \quad \sup \left \{ \mathbb J (\varphi, \psi) : (\varphi, \psi) \in \Phi\_c \right \}, \\
(\mathrm{DP'}) &: \quad \sup \left \{ \mathbb J (\varphi, \psi) : (\varphi, \psi) \in \Phi'\_c \right \}.
\end{align}
$$


Clearly,
$$
\Phi\_c \subset \Phi\_c'
\quad \text{and} \quad
\sup \mathrm{DP} \le \sup \mathrm{DP'} \le \inf \mathrm{KP}.
$$


The central definition leading to the existence of solutions of above problems is $c$-concavity, i.e.,



> 
> **Definition 2.33** A function $\varphi: X \rightarrow \mathbb{R} \cup\{-\infty\}$ is said to be **$c$-concave** if there exists $\psi: Y \to \mathbb{R} \cup\{-\infty\}$ such that $\psi \not \equiv-\infty$ and that
> $$
> \varphi(x) = \psi^c (x) := \inf \_{y \in Y}[c(x, y)-\psi(y)] \quad \forall x \in X.
> $$
> Here $\varphi$ is called the **$c$-conjugate** of $\varphi$.
> 
> 
> 


I'm able to prove that



> 
> **Theorem** Let $c$ be real-valued and lower semi-continuous. Assume there exists $(c\_X, c\_Y) \in L\_1 (\mu) \times L\_1(\nu)$ such that they are real-valued and that $c(x,y) \le c\_X(x) + c\_Y (y)$ for all $(x,y) \in X \times Y$. Then $\mathrm{DP'}$ admits a solution.
> 
> 
> 


At the bottom of page 87 of Villani's **Topics in Optimal Transport**, there is **Exercise 2.36** to prove that $\mathrm{DP'}$ admits a maximizer, i.e.,



> 
> **Exercise 2.36** Let $c$ be lower semi-continuous. Assume there exists $(c\_X, c\_Y) \in L\_1 (\mu) \times L\_1 (\nu)$ that are non-negative such that
> $$c(x,y) \le c\_X (x) + c\_Y(y) \quad \forall (x, y) \in X \times Y.$$
> Then $\mathrm{DP'}$ admits a solution of the form $(\varphi, \varphi^c) \in \Phi\_c'$.
> 
> 
> 


**My attempt:** To make it easier, for **Exercise 2.36** I assumer further that $c, c\_X, c\_Y$ are real-valued. By **Theorem**, $\mathrm{DP'}$ admits a solution $(\varphi, \psi) \in \Phi'\_c$. Now we **assume** that $\varphi^c \in L\_1 (\nu)$. By definition,
$$
\varphi^c (y) \le c(x, y)-\varphi(x) \quad \forall (x, y) \in X \times Y.
$$


So $(\varphi, \varphi^c) \in \Phi'\_c$. Also,
$$
\varphi^c (y) = \inf\_{x \in X} (c(x, y)-\varphi(x)) \ge \inf\_{x \in X} (\psi (y)) = \psi (y).
$$


So $\mathbb J (\varphi, \varphi^c) \ge \mathbb J (\varphi, \psi)$. Then $(\varphi, \varphi^c)$ satisfies the requirement. This is called by the author as *double convexification trick*.




---


**My question:** The first issue is to prove that $\varphi^c$ is measurable, and the second one is to prove that $\varphi^c$ is $\nu$-integrable. However, the measurability of $\varphi^c$ is [subtle and non-trivial](https://mathoverflow.net/questions/434969/optimal-transport-how-is-the-use-of-disintegration-theorem-valid-in-this-constr). Above exercise is exactly **Theorem 4.10** in [this lecture note](https://drive.google.com/file/d/1-6FpiqPG4GVDEBk1qgsGU141oVKyxVG-/view) in which the measurability of $\varphi^c$ is **not** proved.


In optimal transport, this is a fundamental result in both theory and practice. Could you elaborate on how to finish **Exercise 2.36**?


Thank you so much for your elaboration!



Optimal transport: the existence of an optimal pair of $c$-conjugate functions
I have found a related paper [Existence and stability results in the $L^1$ theory of optimal transportation](https://www.math.ucdavis.edu/%7Esaito/data/emd/ambrosio-pratelli.pdf) by Luigi Ambrosio and Aldo Pratelli. Step 2 of the proof of Theorem 3.2 is



> 
> Step 2. Now we show that $\psi:=\varphi^c$ is $\nu$-measurable, real-valued $\nu$-a.e. and that
> $$
> \varphi+\psi=c \text { on } \Gamma \text {. }
> $$
> It suffices to study $\psi$ on $\pi\_Y(\Gamma)$ : indeed, as $\gamma$ is concentrated on $\Gamma$, the Borel set $\pi\_Y(\Gamma)$ has full measure with respect to $\nu=\pi\_{Y \#} \gamma$. For $y \in \pi\_Y(\Gamma)$ we notice that $(10)$ gives
> $$
> \psi(y)=c(x, y)-\varphi(x) \in \mathbb{R} \quad \forall x \in \Gamma\_y:=\{x:(x, y) \in \Gamma\}
> $$
> In order to show that $\psi$ is $\nu$-measurable we use the disintegration $\gamma=\gamma\_y \otimes \nu$ of $\gamma$ with respect to $y$ (see the appendix) and notice that the probability measure $\gamma\_y$ is concentrated on $\Gamma\_y$ for $\nu$-a.e. $y$, therefore
> $$
> \psi(y)=\int\_X c(x, y)-\varphi(x) d \gamma\_y(x) \quad \text { for } \nu \text {-a.e. } y
> $$
> Since $y \mapsto \gamma\_y$ is a Borel measure-valued map we obtain that $\psi$ is $\nu$-measurable.
> 
> 
> 


In the proof, $\psi$ is proved to be $\nu$-measurable. It seems to me being $\nu$-measurable is equivalent to being [Bochner measurable](https://en.wikipedia.org/wiki/Bochner_measurable_function), i.e., $\psi$ is a $\nu$-a.e. pointwise limit of a sequence of simple functions. So $\psi$ is $not$ necessarily Borel measurable.


