Let $X, Y$ be Polish spaces and $c:X \times Y \to [0, +\infty)$ lower semi-continuous. Assume that $f:X \to \mathbb R \cup\{\pm\infty\}$ is measurable. We define the $c$-transform $f^c:Y \to \mathbb R \cup\{\pm\infty\}$ of $f$ by $$ f^c(y) := \inf_{x\in X} [c(x, y)- f(x)] \quad \forall y \in Y. $$ This $c$-transform is fundamental in optimal transport.
Can we prove that $f^c$ is measurable?