Let $F(t,a):T\times A \to \mathbb{R}$, continuous in $a$ for each $t$ and measurable in $t$ for each $a$.
Is there a selection theorem for the case
$$
I(t) = \{a\in A: F(t,a) = B(t)\}
$$
I found a reference where $B$ is a fixed closed subset.
($F(t,a)\in B$ in that case)
However, I have $B(t)$ (single valued) depends on $t$.
(Spaces are as nice as possible.)