B.1
By egorov, $ \forall \ k $ we may pick a set $ E_k $ such that $ m(E_k)<\frac{1}{2^k} $ and $ f_n \rightarrow 0 $ uniformly off of $ E_k $. In particular $ \exists $ a sequence of integers $ {n_k} $ increasing in k such that
$ x \not\in E_k \Rightarrow f_{n_k}(x) < \frac{1}{2^k} $
So for a given $ x, \ \sum_{k=1}^\infty f_{n_k}(x) $ will converge as long as $ \exists \ K $ such that $ x \not\in E_k \ \forall \ k>K. $
In other words, $ \sum_{k=1}^\infty f_{n_k}(x) $ will converge provided $ x \not\in \cap_{K=1}^\infty \cup_{k=K}^\infty E_k=limsup E_k $
But $ \sum_k m(E_k) < \infty \Rightarrow m(limsup E_k)=0 $ by Borel Cantelli (cf Problem Set #4)
$ \Rightarrow \sum_{k=1}^\infty f_{n_k}(x) $ converges for a.e. x.
--Wardbc 12:15, 24 July 2008 (EDT)
