In the book "Optimal Transport for Applied Mathematicians" by Santambrogio (Theorem 1.33), the author makes the following claim:
Let $\gamma$ be a coupling of two probability measures $\mu,\nu$, and $c:X\times Y \to [0,\infty)$ be a cost function. Then, $K(\gamma) = \int c(x,y) d\gamma$ is continuous. Assume that $X\times Y$ is compact.
My question is how does one proves that $K$ is continuous?
Note that we are using weak convergence, hence, $$\gamma_n \to_w \gamma \iff \forall f\in C_b, \int f d\gamma_n = \int f d\gamma$$
Therefore, if cost $c$ is a continuous function, then $K$ is clearly continuous. But I don't know how to prove if $c$ is not continuous. The author doesn’t make clear the assumptions regarding $c$. Other than $c$ continuous, under which other conditions is $K$ continuous?