Homework 3, Real Analysis 2

Due February 23

Probl

1. For any two Cantor sets $latex \mathcal C_1, \mathcal C_2$, as constructed in HW2, Problem 3, there exists a continuous, bijective and increasing function $latex F:[0,1]\to[0,1]$ that maps $latex \mathcal C_1$ surjectively onto $latex \mathcal C_2$.


2. There exists a measurable function f and a continous function $latex \Phi$ such that $latex f\circ\Phi$ is non-measurable.

Problem 2

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

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

Problem 3 (Tchebychev inequality)

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

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

Problem 4

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

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

Problem 5

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