Jtdylktuy

This is part of an exercise from "Real analysis for graduate students" by Richard Bass: Let $m$ be a Lebesgue measure. Suppose for each $n$, $A_n$ is a Lebesgue measurable subset of $[0,1]$. Let $B$ consist of those points $x$ that are in infinitely many of the $A_n$. Show that $B$ is Lebesgue measurable.

I'm in general undecided about what to use to prove that something is Lebesgue measurable. Is it a good way to use outer measure definition and try to show $m^*(E)=m^*(Ecap B)+ m(Ecap B^c)$, for instance. It seemed hard to prove this way for me, and I was wondering what is the best way. Thanks!

Let ${A_n}$ be a countable collection of sets and let $B$ denote the set of elements which are in $A_n$ for infinitely many $n$. Then

$$B = bigcap_{n=1}^{infty}bigcup_{k = n}^{infty}A_k.$$

To see this, note that $x in bigcaplimits_{n=1}^{infty}bigcuplimits_{k = n}^{infty}A_k$ if and only if $x in bigcuplimits_{k=n}^{infty}A_k$ for all $n$. The latter occurs if and only if $x in A_n$ for infinitely many $n$; if $x$ were only in finitely many of the sets, say $A_{n_1}, dots, A_{n_j}$, then $x notin bigcuplimits_{k=n_j+1}^{infty}A_k$.

As the collection of Lebesgue measurable sets is closed under countable unions and countable intersections (it is a $sigma$-algebra), $B$ is Lebesgue measurable.

A similar exercise is to show that the set of elements which belong to $A_n$ for all but finitely many $n$ is measurable. Denoting this set by $C$, you can show that $C$ is Lebesgue measurable by using the fact that

$$C = bigcup_{n=1}^{infty}bigcap_{k = n}^{infty}A_k.$$