Folland Real analysis Exercise 2.4.37

Question

I want to prove Exercise 2.4.37 Modes of Convergence in Folland

Suppose that $f_n$ and $f$ are measurable and $g$ be a function

1: if $g$ is continuous and $f_n \to f$ a.e, then $g \circ f_n \to g \circ f$ a.e

2: if $g$ is uniformly continuous and $f_n \to f$ uniformly, almost uniformly, or in the measure, then $g \circ f_n \to g \circ f$ uniformly, almost uniformly, or in the measure

so if you give reasonable hints, I will be very happy. Thanks!

Answer 1

Let $D$ be the domain of the functions $f$ and $f_n$ and let $N \subseteq D$ be a null-set such that

$f_n(x) \to f(x)$ for all $x \in D \setminus N$. Since $g$ is continuous we have

$g(f_n(x)) \to g(f(x))$ for all $x \in D \setminus N$.

This shows 1.

Are you now in a position to show 2. ?