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Homework 6, Real Analysis 2

Due March 16

Problem 1

Consider the function on $latex \R$ given by

$latex f(x) = \begin{cases}\dfrac{1}{|x|(\log |x|)^2} & |x|\le 1/2\\0 & \text{otherwise.}\end{cases}$

  1. f is integrable.

  2. $latex f^*(x) \ge \dfrac{c}{|x|\log 1/|x|}$ for some $latex c>0$ and all $latex |x|\le 1/2$.

  3. $latex f^*$ is not locally integrable.

Problem 2

Let $latex E\subset [0,1]$ be a measurable set such that there exists $latex \alpha > 0$ such that

$latex m(E\cap I) \ge \alpha m(I)$

for all intervals $latex I\subset[0,1]$. Then $latex m(E)=1$.

Problem 3

Let $latex F\subset\R$ be closed and $latex \delta(x)$ the distance from x to F. Then

$latex \delta(x+y) =o(|y|)$

for almost every $latex x\in F$.

Problem 4

Suppose $latex \{K_\delta\}$ is a family of kernels that satisfies

  • $latex \displaystyle \int_\R K_\delta = 0$ for all $latex \delta>0$.

  • For some $latex A>0$, $latex |K_\delta(x)| \le A \min\{\delta^{-d}, \delta/|x|^{d+1}\}$ for all $latex \delta > 0, x\in\R^d$.

If $latex f\in L^1(\R)$, then

$latex f*K_\delta(x) \to 0$

for a.e. x as $latex \delta\to 0$.