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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