Let $A$ be a finite subset and $B$ be a compact subset of $\mathbb{R}^n$, and $\Omega$ is an open bounded domain in $\mathbb{R}^n$. Let $f:\Omega\times A\times B\to \mathbb{R}$ be a Caratheodory function, in the sense that it is Lebesuge measurable in the first component, and continuous in the last two components.
Suppose that $\alpha:\Omega\to A$ and $\beta:\Omega \to B$ are Lebesuge measurable functions, such that we have for a.e. $x\in \Omega$ that, $$ f(x,\alpha(x),\beta(x))=\max_{b\in B}f(x,\alpha(x),b). $$ I was wondering whether we can find a Lebesuge measurable function $\tilde{\beta}:\Omega\times A\to B$, such that $$ \beta(x)=\tilde{\beta}(x,\alpha(x)),\quad \textrm{and}, \quad f(x,a,\tilde{\beta}(x,a))=\max_{b\in B}f(x,a,b). $$ for a.e. $x\in \Omega$ and all $a\in A$?
I feel that it looks like an implicit theorem, or a measurable extension theorem. But I am quite new to these measurable selection theorems. So I would really appreciate if you could point out the correct reference.