Thank you, this is very helpful

Thanks for helping. Let us say: $E = C([-1,1]$, with the sup norm $\| \cdot \|_\infty$, $K = [0,T] \subset \mathbb{R}$, and $C(K,E)$ has now norm $\| u \| = \sup_{t \in [0,T]} \| u(t) \|_\infty$. And I am now considering $f \colon \Omega \to C(K,E)$, or $f \colon \Omega \to C([0,T],C([-1,1]))$ to spell it out

Also, this is a research question, really: the function $f$ I am considering comes from the solution to a Banach-valued dynamical system, and I am trying to understand how to prove measurability for the particular $f$ that I am after

@ThomasKojar thanks, I have now changed the question so as to explain that $C(K,E)$ is the space of continuous functions on $K$ to $E$, with standard norm.