Homework 1, Real Analysis 2

Due February 9

Problem 1

The Cantor set is totally disconnected (for any x\not= y\in C there is z\not\in C between x and y) and perfect (compact and without isolated points).

Problem 2

Let E\subset \R and

O_n = \{x: d(x,E) < 1/n\}.

  1. If E is compact, m(E) = \lim_{n\to\infty} m(O_n)
  2. The previous conclusion may be false for closed and unbounded, or open and bounded.

Problem 3 (Borel-Cantelli lemma)

Let \{E_k\} be a sequence of measurable sets such that

\displaystyle \sum_{k=1}^\infty m(E_k) < \infty,

and define E = \limsup_{k\to\infty} E_k = \{x\in\R^d: x\in E_k for infinitely many k\}.

  1. E is measurable.
  2. m(E) = 0.

Problem 4

Let, for a subset E\subset\R^d,

\displaystyle m_*^R(E) = \inf \sum_{j=1}^\infty |R_j|,

where the infimum is taken over all countable covers \{R_j\} for E of rectangles. Then

m_*^R(E) = m_*(E).

We can thus conclude that we obtain the same Lebesgue measure theory if cubes are replaced by rectangles.

Post Tagged with 

Comments & Responses

Leave a Reply