Real Variables: Measurability of {x: x∈An i.o.}

[glacier302]

Homework Statement

Let An, n = 1,2,..., be a sequence of measurable sets. Let E = {x: x∈An i.o.}.

(a) Prove that E is a measurable set.

(b) Prove that m(E) = 0 if ∑m(An) < ∞

Homework Equations

A point x is said to be in An infinitely often (i.o.) if there is an infinite sequence of integers n1<n2<... such that x∈Ank for every k.

The Attempt at a Solution

I'm really not sure where to start with part (a). For part (b), if ∑m(An) < ∞
then E is countable, therefore m(E) = 0...I can't really explain why E is countable, though, it's just an instinct.

Any hints would be greatly appreciated : )

 

[CompuChip]

My first instinct would be to try and explicitly define the measure on E.
Each $A_n$ comes with its own measure $\mu_n$ so you could try something like
$$\mu_E(x) := \sum_{n \mid x \in A_n} \mu_n(x)$$
and check that it is a measure.

 

[glacier302]

I think I figured it out. Thank you!