This is Exercise 1.5.7 from Tao's An introduction to measure theory.I've proved (1),but I don't know how to solve (2).
Exercise 1.5.7.Let $(X,B,\mu)$ be a measure space,let $f_n:X→C$ be a sequence of measurable functions converging pointwise almost everywhere as n→$\infty$ to a measurble limit$f:X→C$,and for each n,let$f_{n,m}:X→C$ be a sequence of measurable functions converging pointwise almost everywhere as $m→\infty$(keeping n fixed) to $f_n$.
(1) If $\mu(X)$ is finite,show that there exists a sequence $m_1,m_2,...$ such that $f_{n,m_n}$ converges pointwise almost everywhere to $f$.
(2) Show the same claim is true if,instead of assuming that $\mu(X)$ is finite,we merely assume that X is $\sigma-finite$,i.e.it is the countable union of sets of finite measure.
Any help is appreciated.