Let $f \in L^p(0,T;X)$ where $X$ is a separable Banach space.
So $f$ is a Bochner function and hence Bochner measurable, meaning that there is a sequence of measurable countably-valued functions that converges to $f$ a.e in $t$.
Let us denote $([0,T],S)$ as the measure space associated to the time interval. How do I know that $u:[0,T] \to X$ is $S$-measurable? I don't know what sigma-algebra $S$ should be though.