Fix $\Omega \subseteq \mathbb{R}^N$ a bounded domain (of whatever smoothness you end up needing, let's say $C^1$ domain for definiteness) and fix some $0 <T < \infty$. In considering evolution equations in PDE, we may concern ourselves with spaces of functions $[0,T] \to X$ where $X$ is a Banach space of functions $\Omega \to \mathbb{R}$ ($X = L^p(\Omega), W^{k,p}(\Omega)$, etc.). For functions of this form, we can introduce notions of measurability and weak (time) derivatives in connection with the Bochner integral (e.g. as in Evans PDE).

But also to any function $\textbf{f} : [0,T] \to X$ we can associate a function $f : [0,T] \times \Omega \to \mathbb{R}$ in the obvious way: $f(t,x) = \textbf{f}(t)(x)$. Then $f$ is a function defined on a subset of $\mathbb{R}^{N+1}$ and so has natural notions of measurability and weak time partial derivatives. This suggests the fairly natural questions: is $f$ measurable iff $\textbf{f}$ is (note that if $X$ isn't separable, there are a few different notions of measurability to consider for $\textbf{f}$)? And is $\textbf{f}$ weakly differentiable in time iff $\partial_t f$ exists in the weak sense?

More generically, the question is how can the highly related notions defined separately for function-space-valued functions and for functions on $\mathbb{R}^{N+1}$ be connected? It seems like we'd really want them to agree, at least in nice cases, but I'm not quite convinced they will.

The book I've looked at (Evans, Wloka) haven't touched on these seemingly natural questions, so any pointers to resources that cover this sort of thing would also be useful.