Understaning a problem involving measurable sets of a bounded sequence of measures Hi guys I am trying to understand my problem,
We have a sequence of measureable sets $\{A_n\}$ Where $\sum _{n=1} ^{\infty}m(A_n) < \infty$
Define a set $B= \{x \in \mathbb  R: \# \{ n:x \in A_n \}=\infty \}$
We want to show m(B)=0
My question is what exactly is B. How I am understanding it is the set of x such that the number of x in $E_n$ is infinity many. That does not quite make sense 

Let $f_i = \chi_{A_i}$ be the characteristic function of $A_i$ and $f = \sum_{i=1}^\infty f_i $. Then 
$$\int f d\mu = \sum_{i=1}^\infty m(A_n) < \infty $$
(The equality can checked using, e.g., monotonce convergence theorem) Thus $f$ is integable. Thus it is finite a.e.. This is what we want as $B = \{ f= \infty\}$. 
