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Homework 3, Real Analysis

Due Septembre 7

Problem 1

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

Problem 2

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

  1. Prove that it is sufficient to assume that E is compact, and thus $latex |E|_* < \infty.$

  2. Given $latex \e>0$, choose an open $latex U\supset E$ with $latex |U|_* < |E|_* + \e.$ Prove that we can write $latex U\setminus E = \bigcup I_j$, where the $latex I_j$ are disjoint open intervals.

  3. If $latex I,J$ are disjoint open intervals, then $latex |I\cup J|_* = |I| + |J|.$

  4. For each N, $latex |U|_* \ge |E|_* + \sum_{j=1}^N |I_j|$.

  5. Conclude $latex |U\setminus E|_* < \e$.

Problem 3

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

Problem 4

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

  1. If E is compact, then $latex |E| = \lim |U_n|$.

  2. The previous may fail is E is closed and unbounded, or bounded and open.