The following is a problem from Stei-Shakarchi's Real Analysis:
Suppose $(E_n)$ be a countable family of measurable sets such that $\sum_n m(E_n)<\infty$. Define $E=\{ x\in\mathbb{R}^d\colon x\in E_k \mbox {for infinitely many } k\}$. The problem is to show that $E$ is measurable. The hint given is write $E=\cap_{n=1}^{\infty} \cup_{k\geq n} E_k$.
I could not understand the Hint. I mean, from given definition of $E$, I couldn't come to the interpretation in hint. Can you help me?
Explanation of the definition:
We'd like to somehow describe the set of elements that appear in $E_n$ for infinitely many $n \in \Bbb N$.
Let's, however, start bigger. What if we were to describe the elements that appear in some $E_k$ for all $k \geq 1$? That would be the union of all $E_k$, or more simply $$ A_1 = \bigcup_{k \geq 1} E_k $$ What about the elements that appear in some $E_k$ for some $k \geq n$? That would, by the same idea, be $$ A_n = \bigcup_{k \geq n} E_k $$ It should be clear that $A_1 \supseteq A_2 \supseteq \cdots \supseteq A_n \supseteq A_{n+1} \supseteq \cdots$. Now, which elements appear in every $A_n$? Well, if $x \in A_n$ for every $n$, then we must be able to say that for all $n$, there is a $k \geq n$ such that $x \in E_k$. You should also be able to see that any element $x$ satisfying those conditions is an element of every $A_n$.
Equivalently, we could say that $x \in \bigcap_{n=1}^\infty A_n$ if and only if $x \in E_n$ for infinitely many $n$. That is, we can write the set of elements that appear in infinitely many $E_n$ as $$ \limsup_{n \to \infty} E_n = \bigcap_{n=1}^\infty A_n = \bigcap_{n=1}^\infty \left(\bigcup_{n\geq k} E_k\right) $$ as the hint claims.
As for applying the hint: show that each $A_n$ is measurable. By the nature of measurability, it follows that $\bigcap_{n = 1}^\infty A_n$, the countable intersection of measurable sets, is measurable.