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 $\varphi^c = \lim \_{\ell \rightarrow \infty} \psi\_{\ell}$?
> 
> 
> 




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I posted [this question](https://math.stackexchange.com/questions/4579160/how-can-varphic-be-written-as-varphic-lim-ell-rightarrow-infty) on MSE, but have not received any answer so far. So I post it here.



Optimal transport: how $\varphi^c$ can be written as $\varphi^c = \lim _{\ell \rightarrow \infty} \psi_{\ell}$?
Are you sure there aren’t additional conditions on $\varphi$? Because otherwise taking $X = \mathbb R$ and $Y$ to be a one point space, the following gives a counterexample:


$c(x, y) = 0$ if $x = 0$; $c(x, y) = 1$ otherwise, and


$\varphi(x) = 0$ if $x = 0$, $\varphi(x)= 2$ otherwise.


Indeed, $\varphi^c = -1$, while $\psi\_\ell \leq -2$ for all $\ell$.


My guess is that $\varphi$ should be restricted to be continuous.


