Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$].


For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, we consider the *$c$-transform*
$$\tag{1}
T\_c \, : \, L^1(\mu) \,\ni\, \varphi \,\mapsto\, \varphi^c, \quad\text{where}\quad \varphi^c(y):= \inf\_{x\in X} c(x,y) - \varphi(x)
$$


(as known e.g. from the theory of optimal transport). Let $\varphi\in L^1(\mu)$ and $(\varphi\_\alpha)\subset L^1(\mu)$ be a net.


My **question** is: If $\varphi, (\varphi\_\alpha)\in C\_b(X)$ with $\varphi\_\alpha\rightarrow\varphi$ uniformly on compact sets, does this imply that $T\_c(\varphi\_\alpha)\rightarrow T\_c(\varphi)$ in $L^1(\nu)$ ?


Any hints or references are welcome.



Continuity of generalised Legendre transform
$\newcommand\vpi\varphi$The answer is no. E.g., suppose that $X=Y=\mathbb R$, $c=0$, and $\vpi\_a(x)=\min(a,\max(0,x-a))$ for $a\ge0$ and all real $x$. Then, as $a\to\infty$, we have $\vpi\_a\to0$ uniformly on compact sets, whereas $T\_c(\vpi\_a))=-a\not\to0=T\_c(0)$ in $L^1(\nu)$ for any probability measure $\nu$ over $\mathbb R$.


