Let $E$ be a measurable subset of $\mathbb{R}$ and let $F_n :E \to \mathbb{ R} $ be a sequence of measurable functions. Let $$E^*= \{ x\in E : F_n \text{ is convergent } \}$$ Is the set $E^*$ measurable?
To solve this, I need to understand the relation between the set $E^*$ and the functions $F_n$. They are strongly related, but how can I express this relation in terms of sets and measure?
I preassume that $E^{*}:=\{x\in E: (F_n(x))_n\text{ is a convergent sequence}\}$ and mention that $\mathbb N$ denotes the set of positive integers in this answer.
Then for $x\in E$:
$$x\in E^{*}\iff\forall k\in\mathbb{N}\exists n\in\mathbb{N}\forall r,s\in\mathbb{N}\left[r,s\geq n\implies\left|F_{r}\left(x\right)-F_{s}\left(x\right)\right|<\frac{1}{k}\right]$$ So if $\mathbb N_n$ denotes $\{n,n+1,\dots\}$ then:
$$E^{*}=\bigcap_{k\in\mathbb{N}}\bigcup_{n\in\mathbb{N}}\bigcap_{\left(r,s\right)\in\mathbb{N_n}^{2}}\left\{ x\in E:\left|F_{r}\left(x\right)-F_{s}\left(x\right)\right|<\frac{1}{k}\right\}$$
The sets $\left\{ x\in E:\left|F_{r}\left(x\right)-F_{s}\left(x\right)\right|<\frac{1}{k}\right\} $ are measurable and we are dealing with countable unions and intersections.
So $E^{*}$ is measurable.
