In Giovanni's book, A First Course in Sobolev Spaces, he says
Theorem 8.28. Let $I \subset \mathbb{R}$ be an open interval, let $E \subset \mathbb{R}^n$ be a Lebegue measurable set, and let $1 \le p < \infty$. Then $L^p(I; L^p(E))$ can be identificed with $L^p(E \times I)$.
I understand most of the proof, except that given $u \in L^p(E \times I)$, $v(t) := u(\cdot\,; t)$ is a strongly measurable in $L^p(E \times I)$. Since he defines $T: L^p(E \times I) \to L^p(I; L^p(E))$ with $T(u) = v$, it would not make sense if $v$ is not strongly measurable.
I tried to define a sequence of simple functions in $L^p(I; L^p(E))$ to approximate $v$, but no luck. I'd appreciate any help! Thank you.