Homework 3, Real Analysis 2

Due February 23

Probl

  1. For any two Cantor sets \mathcal C_1, \mathcal C_2, as constructed in HW2, Problem 3, there exists a continuous, bijective and increasing function F:[0,1]\to[0,1] that maps \mathcal C_1 surjectively onto \mathcal C_2.
  2. There exists a measurable function f and a continous function \Phi such that f\circ\Phi is non-measurable.

Problem 2

Given a collection of sets E_1, E_2, \ldots, E_n, there exists a disjoint collection F_1, F_2, \ldots, F_N, N=2^n - 1, such that \bigcup E_j = \bigcup F_k and each

\displaystyle E_j = \bigcup_{F_k\subset E_j} F_k.

Problem 3 (Tchebychev inequality)

Let f\ge0 be integrable, \alpha > 0 and E_\alpha = \{x:f(x)>\alpha\}. Then

\displaystyle m(E_\alpha) \le \frac{1}{\alpha}\int f.

Problem 4

Let f\ge 0 be finite almost everywhere, E_k = \{ x:f(x) > 2^k\} and F_k = \{x: 2^k < f(x) \le 2^{k+1}\}. Then f is integrable if and only if

\displaystyle \sum_{k=-\infty}^\infty 2^k m(E_k) < \infty and \displaystyle \sum_{k=-\infty}^\infty 2^k m(F_k) < \infty.

Problem 5

Let f(x) = |x|^{-a}\chi_{|x|\le 1} and g(x) = |x|^{-b}\chi_{|x|>1} on \R^d. Then f is integrable if and only if a<d, and g is integrable if and only if b>d.

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