Banach Spaces: Pointwise Limit vs. Uniform Limit

Question

Agreement

All notions are up to null sets.

Limits are meant by simple functions.

Problem

Given a finite measure space $\mu(\Omega)<\infty$ and a Banach space $E$.

Consider bounded measurable functions $F:\Omega\to E: \|F\|_\infty<\infty$.

Precisely the pointwise limits have separable image: $$S_n\stackrel{0}{\to}F\iff \mathrm{im}F\text{ separable}$$ and precisely the uniform limits have precompact image: $$S_n\stackrel{\infty}{\to}F\iff \mathrm{im}F\text{ precompact}$$ (Is it right like this?)

Now, what is an example of a pointwise limit but not a uniform limit? $$S_n\stackrel{0}{\to}F,\,S'_n\stackrel{\infty}{\nrightarrow}F$$

Note that the special ingredients are: Finite Measure + Bounded & Measurable Function

Answer 1

And again a variant of the famous example: $$F:(0,1]\to\mathcal{H}:F(\frac{1}{n+1}<t\leq\frac{1}{n}):=e_n$$ Clearly, it is pointwise limit but can't be uniform limit.