I have just encountered the notion of a measure space. I am trying to solve the following problem but I don't really know how to start it:
Let $A_n \in \frak{X}$ where $(X, \frak{X}, \mu)$ is a measure space. Show $\limsup_{n \to \infty} \chi_{A_n}(x) = \chi_A(x)$, where $A = \bigcap_{m=1}^{\infty} \bigcup_{n \geq m}A_n$
Further, if $\sum_{n=1}^{\infty}\mu(A_n) < \infty$ show $\mu(A) = 0$
Obviously for the first part I need to somehow use the properties of the $\sigma$-algebra, but I don't see where they come into play. Also, I'm not sure whether the two parts of the question are related or not. Using the definition of $\limsup$: $\limsup \chi_{A_n}(x) = \lim_{m \to \infty} \sup\{\chi_{A_n}(x) | n \geq m \}$ $ = \lim_{m \to \infty} \chi_{B_m}(x),$
where $B_m = \bigcup_{n \geq m}A_n$.
I'm not sure what to do from here. Is it clear that $\lim_{m \to \infty} \chi_{B_m}(x) = \chi_A(x)$?