TransWikia.com

Show that $m({xin[0,1]:text{$x$ lies in infinitely many $E_j$}})geqfrac{1}{2}$ when $m(E_j)geqfrac{1}{2}$

Mathematics Asked on December 23, 2021

Question: Suppose $E_1, E_2,ldots$ is a sequence of measurable subsets of $[0,1]$ with $m(E_j)geqdfrac{1}{2}$. Show that $m({xin[0,1]:text{$x$ lies in infinitely many $E_j$}})geqdfrac{1}{2}$, where $m$ is the one dimensional Lebesgue measure.

My thoughts: I would imagine that one could use the Borel-Cantelli Lemma here. The issue is that for the Borel Cantelli Lemma I need $sum_{j=1}^infty m(E_j)<infty$, but since $m(E_j)geqfrac{1}{2}$, then I can’t use it. Now, if the hypotheses of the Borel Cantelli Lemma were satisfied, then $m(cap_{n=1}^inftycup_{n=k}^infty E_j)=0$. Which, to my understanding, is the same things as saying that if $E={xin[0,1]:text{$x$ lies in infinitely many $E_j$}}implies m(E)=0$, where $E$ is the set we are dealing with in the question. So, I am wondering if there is a general way to sort of "shift" the Borel Cantelli Lemma. Or, is there another way to go about this problem? Any thoughts, ideas, answers, etc. are always greatly appreciated! Thank you.

One Answer

Let $F_j=[0,1]setminus E_j$. Then $m (F_j ) leq frac 1 2$ and , by Fatou's Lemma, $int lim inf I_{F_j} dm leq lim inf m(F_j) leq frac 1 2$ Can you finish ?

[ Note that $lim inf I_{F_j}$ is the indicator function of $lim inf F_j$].

Answered by Kavi Rama Murthy on December 23, 2021

Add your own answers!

Ask a Question

Get help from others!

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP