# Homework 3, Real Analysis

## Due Septembre 7

### Problem 1

If $E, F\subset\R$ are a closed and a compact disjoint sets, then $\text{dist}(E,F)>0$.

### Problem 2

If $E\subset\R$ is closed, then it is measurable. Follow the next steps.

1. Prove that it is sufficient to assume that E is compact, and thus $|E|_* < \infty.$
2. Given $\e>0$, choose an open $U\supset E$ with $|U|_* < |E|_* + \e.$ Prove that we can write $U\setminus E = \bigcup I_j$, where the $I_j$ are disjoint open intervals.
3. If $I,J$ are disjoint open intervals, then $|I\cup J|_* = |I| + |J|.$
4. For each N, $|U|_* \ge |E|_* + \sum_{j=1}^N |I_j|$.
5. Conclude $|U\setminus E|_* < \e$.

### Problem 3

Find a sequence of measurable sets $E_1 \supset E_2 \supset ...$ such that, for $E = \bigcap E_j$, $|E| \not= \lim |E_j|$.

### Problem 4

For $E\subset\R$, let $U_n=\{x\in\R: \text{dist}(x,E) < 1/n\}$.

1. If E is compact, then $|E| = \lim |U_n|$.
2. The previous may fail is E is closed and unbounded, or bounded and open.
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