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

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 2.33A function $\varphi: X \rightarrow \mathbb{R} \cup\{-\infty\}$ is said to be$c$-concaveif 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$c$-conjugateof $\varphi$.

I'm able to prove that

TheoremLet $c$ 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 2.36Let $c$ 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. Above exercise is exactly **Theorem 4.10** in this lecture note 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!