Let $X = Y = \mathbb{R}^d$ and let $\nu$ be a probability measure on $\mathbb{R}^d$. Consider the collection of probability measure $\pi$ on $X\times Y$ such that $\pi$ has $y$-marginal $\nu$:

$$ \Pi(\nu) = \{\pi: \pi(X,dy) = \nu(dy)\}. $$

Let $f:X\times Y \mapsto \mathbb{R}$ be a measurable function (you can assume it is continuous and has nice growth condition) such that the partial minimization

$$ \phi(y) = \inf\{f(x,y):x\in X\} $$ is measurable. My question: is it true that

$$ \inf_{\pi \in \Pi(\nu)} \int f(x,y) d\pi = \int \phi(y) d\nu, $$ and if so, can it be proved without using any sort of measurable selection?