# Riesz-Fischer in \$L^infty\$ clarification needed.

This is Rudin’s RCA book. He discusses completeness of $$L^infty(mu)$$, but there is a part of proof that I don’t understand.

In $$L^infty(mu)$$, suppose $${f_n}$$ is a Cauchy sequence, and let $$A_k, B_{m,n}$$ be the sets where $$|f_k(x)| > |f_k|_infty$$ and $$f_n(x)-f_m(x)| > |f_n-f_m|_infty$$, and let $$E$$ be the union of these sets, for $$k, m ,n = 1,2,3, cdots$$. Then, $$mu(E)=0$$, and on the completement of $$E$$, the sequence $${f_n}$$ converges uniformly to a bounded function $$f$$. Define $$f(x) = 0$$ on $$E$$. Then, $$f in L^infty(mu)$$ and $$|f_n – f|_infty rightarrow 0$$ as $$n rightarrow infty$$.

Since $$f$$ is a complex function, I understand that $$f_k$$ converges to some measurable complex function $$f$$. The part that I do not understand is 1. (almost everywhere) boundedness, and 2. (a.e.) uniform convergence. Any help would be appreciated.