let $\xi,\eta: \Omega \to \mathbb R$ be i.i.d. random variables on a measurable space $(\Omega , \mathcal F,\mathbb P)$, and let $f: \mathbb R^2 \to \mathbb R$ be a bivariate measurable function (say under Borel $\sigma$-algebra). Clearly, $\omega \mapsto f(\xi(\omega),\eta(\omega))$ is also a random variable.

In many books the authors treat equations like $\inf_{x \in \mathbb R} f(x , \eta)$ as random variables without further explaination about measurability. However, this is equivalent to \begin{equation}g: \omega \mapsto \inf \bigg\{ f(x,\eta(\omega)) \bigg| x \in \mathbb R\bigg\}\end{equation} which is an uncountable infimum of random variables.

Generally speaking, taking uncountable infimum of random variables is not valid. A counter example is shown in Uncountable infimum of measurable functions. The counter example works here if $x$ is subscript instead of a dimension, i.e. $g(w) = \inf_{x\in \mathbb R} f_x(\eta(\omega))$. However, since $f$ is a bivariate measurable function, it seems its measurability prevents the construction in that counter example being valid.

I believe $g$ is always measurable, but I can not come up with a proof. I wonder if there's a clear proof (or even better, a generalization) of this claim.