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 

Note $x\in B$ if and only if for every positive integer $N$, there exists $n \ge N$ such that $x\in A_n$. So $$B = \bigcap_{N = 1}^\infty \bigcup_{n = N}^\infty A_n.$$ Now the sequence $\langle\cup_{n = N}^\infty A_n\rangle_{N = 1}^\infty$ decreases to $B$ and $m(\cup_{n = 1}^\infty A_n) < \infty$ (since $m(\cup_{n = 1}^\infty A_n) \le \sum_{n = 1}^\infty m(A_n) < \infty$), so by continuity of $m$, 
$$m(B) = \lim_{N\to \infty} m\left(\bigcup_{n = N}^\infty A_n\right) \le \lim_{N\to \infty} \sum_{n = N}^\infty m(A_n) = 0.$$
