Sequence of leesgue measurable sets

Let $\{E_n\}$ be a sequence of lebesgue measurable subsets of R such that

$\sum\limits_{n=1}^{\infty} m(E_n) < \infty$ . If $E= \lim_{n} \sup E_{n}$ , then show that m(E)=0

Hello,

Look at the Borel-Cantelli theorem for measure theory.

Quote:

Originally Posted by Chandru1

Let $\{E_n\}$ be a sequence of lebesgue measurable subsets of R such that

$\sum\limits_{n=1}^{\infty} m(E_n) < \infty$ . If $E= \lim_{n} \sup E_{n}$ , then show that m(E)=0

Since $\sum\limits_{n=1}^{\infty} m(E_n) < \infty$ , then for all $\epsilon$ > 0, there exists an N such that if k > N, then $\sum\limits_{n=k}^{\infty} m(E_n) < \epsilon$ . Thus, it must be the case that for each k > N we have $m(E_k) < \epsilon$ . And finally, we have that sup{ $m(E_k)$ } < $\epsilon$ for all k > N.