Thread: Sequence of leesgue measurable sets

Let be a sequence of lebesgue measurable subsets of R such that

. If , then show that m(E)=0

Hello,

Look at the Borel-Cantelli theorem for measure theory.

Originally Posted by Chandru1

Let be a sequence of lebesgue measurable subsets of R such that

. If , then show that m(E)=0

Since , then for all > 0, there exists an N such that if k > N, then . Thus, it must be the case that for each k > N we have . And finally, we have that sup{ } < for all k > N.