Question: Suppose $E_1, E_2,\ldots$ is a sequence of measurable subsets of $[0,1]$ with $m(E_j)\geq\dfrac{1}{2}$. Show that $m(\{x\in[0,1]:\text{$x$ lies in infinitely many $E_j$}\})\geq\dfrac{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 $m(E_j)\geq\frac{1}{2}$, then I can't use it. Now, if the hypotheses of the Borel Cantelli Lemma were satisfied, then $m(\cap_{n=1}^\infty\cup_{n=k}^\infty E_j)=0$. Which, to my understanding, is the same things as saying that if $E=\{x\in[0,1]:\text{$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.
Best 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 $\lim \inf F_j$].