In preparation for the real analysis qualifying exam at my grad school, I've been working through the recommended textbook Modern Real Analysis, by Zimmer (it's free online here).
For the last few days, I've been trying to figure out problem 4.27 (on p124 of the text, which is p132 of the pdf), which asks us to prove the following:
Theorem. If $\{E_k\}_{k=1}^\infty$ is a sequence of Lebesgue measurable subsets of a compact set $K\subseteq \mathbb{R}^n$ such that $\inf_{k\geq 1} \lambda(E_k)>0$, then there's some point which belongs to infinitely many $E_k$'s (i.e. there's some point which belongs to $E_k$ for infinitely many $k$).
Definitions, Etc.
- Lebesgue measure is denoted by $\lambda$.
- The Lebesgue outer measure $\lambda^*$ is defined to be $$\lambda^*(A):= \inf\left\{\sum_{I\in\mathcal{S}} \mathrm{vol}(I) \right\},$$ where the infimum is taken over countable all covers $\mathcal{S}$ of $A$ by sets of the form $[a_1,b_1]\times \cdots \times [a_n,b_n]$ (which Zimmer calls closed $n$-dimensional intervals, or just closed intervals if the dimension is clear).
- A set $A\subseteq\mathbb{R}^n$ is defined to be Lebesgue measurable if $$\lambda^*(A)=\lambda^*(A-S)+\lambda^*(A\cap S)$$ for any set $S\subseteq \mathbb{R}^n$.
My Attempt
If there are finitely many distinct $E_k$'s, this is trivial. So, assume there are infinitely many distinct $E_k$'s (I don't know if this assumption will be useful, but since we're working with compactness, I figured it might). From each $E_k$, choose a point $x_k$. Then because $K$ is compact, there's a subsequence $\{x_{k_j}\}_{j=1}^\infty$ of $\{x_k\}_{k=1}^\infty$ which converges to some point $x\in K$. At this point, I'm stuck. My gut tells me that this point $x$ (or some similarly constructed point) should be in infinitely many $E_k$'s, but I can't for the life of me figure out how to prove this.
Could someone point me in the right direction?