Let $\{E_k\}$ be a sequence of measurable sets in X. Show that $A\in\mathfrak{M}$, where $A$ is the set of all x that lie in infinitely many $E_k$'s.

Question

The problem arises from Walter Rudin's Real & Complex Analysis:

The highlighted texts, however, would NOT make any sense if $A$ is not measurable. All my attempts to show that $A\in\mathfrak{M}$ have failed; any hints would be greatly appreciated.

write $A$ with the $(E_k)$. Hint: an infinite set of $\mathbb N$ is not bounded — blamethelag, May 27 '22 at 12:20
https://math.stackexchange.com/questions/107931/lim-sup-and-lim-inf-of-sequence-of-sets — , May 27 '22 at 12:21

score 1 · Accepted Answer · answered May 27 '22 at 12:26

1

$$ A =\bigcap_{N=1}^\infty \bigcup_{n\ge N} E_k $$

By definition, this means any point of $A$ belongs to infinitely many of $E_k$. By the fact that $E_k$ are all measurable, you conclude $A$ is as well.

answered May 27 '22 at 12:26

Son Gohan

4,397

Let $\{E_k\}$ be a sequence of measurable sets in X. Show that $A\in\mathfrak{M}$, where $A$ is the set of all x that lie in infinitely many $E_k$'s.

1 Answers1