Let $X$ be a separable Banach space. Suppose that $f:[0,T] \times X \to \mathbb{R}$ is such that $t \mapsto f(t,x)$ is measurable.
Is the function $$t \mapsto \sup_{x \in X}f(t,x)$$ also measurable?
I saw this question Supremum of measurable function but I didn't understand the answer.. why we can replace teh supremum over $X$ by a supremum over a ball of size 1.
In Supremum of measurable function there is a specific $f$ with property $f(t, x) = f(t, x/\|x\|)$ for any $x\ne 0$. Thus, you can assume $x$ to be on the $1$-sphere. That isn't valid for any $f$.