Let $\{f_{n}, n \in \mathbb{N} \}$ be a sequence of Lebesgue measurable functions on $[0,1]$.
a) Prove that $f(x) = \sup \{f_{n}(x), n \in \mathbb{N} \}$ is measurable.
b) Let $E \subset [0,1]$ be the set of points where $\{ f_{n} \}$ converges. Prove that E is measurable.
I understand part a)
$\{x \in E : f(x) > c \} = \bigcup_{n=1}^{\infty} \{x \in E : f_n(x) > c \}$ which is measurable as the union of measurable sets.
Part b) is not so straightforward for me, but here is what I have:
If $E=\emptyset$ then $m(E) = 0$ and $E$ is measurable. If $E$ is non-empty and $\{f_{n}\}$ converges pointwise to some point on $[0,1]$, then by Egoroff's theorem $\{f_{n}\}$ converges uniformly on a closed set $F \subset [0,1]$. It is clear that $F = E$ and $E$ is measurable.
I think I conceptually understand part b), but something feels off about my proof.
Thanks for your help!
