Let $X,Y$ be Polish spaces and $c:X \times Y \to [0, \infty]$ lower semi-continuous. There is a sequence $(c_\ell)_{\ell \in \mathbb N}$ with $c_\ell:X \times Y \to [0, \infty)$ of bounded Lipschitz continuous functions such that $c_\ell \uparrow c$ pointwise. Fix $\varphi:X \to \mathbb R$. The following is taken from page 33 of Villani's Topics in Optimal Transportation.
Remark 1.12 (c-concave functions). It follows from the proof that, when $c$ is bounded, one can restrict the supremum in the right-hand side of (1.4) to those pairs $\left(\varphi^{c c}, \varphi^c\right)$, where $\varphi$ is bounded and $$ (1.18) \quad \varphi^c(y)=\inf _{x \in X}[c(x, y)-\varphi(x)], \quad \varphi^{c c}(x)=\inf _{y \in Y}\left[c(x, y)-\varphi^c(y)\right] $$ An easy argument shows that $\left(\varphi^{c c}\right)^c=\varphi^c$ (see Exercise 2.35). The pair $\left(\varphi^{c c}, \varphi^c\right)$ is called a pair of conjugate $c$-concave functions. Note that $\varphi^c$ is measurable, since it can be written (exercise) as $\lim _{\ell \rightarrow \infty} \psi_{\ell}$, where $$ \psi_{\ell}(y)=\inf _{x \in X}\left[c_{\ell}(x, y)-\varphi(x)\right], $$ and $c_{\ell}$ is an increasing family of bounded uniformly continuous functions converging pointwise to $c$. Indeed, each $\psi_{\ell}$ is uniformly continuous, and therefore $\varphi^c$ is measurable. Similarly, $\varphi^{c c}$ is measurable.
I proved that $\psi_\ell$ is bounded Lipschitz continuous for each $\ell \in \mathbb N$.
Could you explain how $\varphi^c$ can be written as $\varphi^c = \lim _{\ell \rightarrow \infty} \psi_{\ell}$?
I posted this question on MSE, but have not received any answer so far. So I post it here.
