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Homework 2, Real Analysis 2

Due February 16

Problem 1

Let $latex E\subset\R$ with $latex m_*(E)>0$.

  1. For each $latex 0 < \alpha < 1$, there exists an open interval I such that $latex m_*(E\cap I) \ge \alpha |I|$.

  2. If E is measurable, the difference set $latex \{x\in\R: x=a-b\text{ for some }a,b\in E\}$ contains an open interval centered at the origin.

Problem 2

Let C be the Cantor set.

  1. $latex x\in C$ if and only if $latex \displaystyle x=\sum_{k=1}^\infty \frac{a_k}{3^k}$, where $latex a_k=0\text{ or } 2$.

  2. The Cantor-Lebesgue function is defined on C by $latex \displaystyle F(x)=\sum_{k=1}^\infty \frac{a_k}{2^{k+1}}$, if $latex \displaystyle x=\sum_{k=1}^\infty \frac{a_k}{3^k}.$ F is well-defined, continuous on C, $latex F(0)=0$ and $latex F(1)=1$, and surjective.

Problem 3

Construct a compact set D in an analogous way as the Cantor set but, at the kth stage of the construction, we remove $latex 2^{k-1}$ central open intervals of length $latex l_k$, with

$latex l_1 + 2l_2 + 4l_3 + \ldots + 2^{k-1}l_k < 1.$

  1. If $latex \displaystyle \sum_{k=1}^\infty 2^{k-1}l_k < 1$, then $latex m(D)>0$.

  2. For each $latex x\in D$, there exists a sequence $latex x_n\not\in D$ such that $latex x_n\to x$ and $latex x_n\in I_n$, where each $latex I_n$ is one of the removed open intervals in the construction of D, with $latex |I_n|\to 0$.

  3. D is perfect, uncountable, and does not contain any interval.

Problem 4

Let $latex \{ f_k\}$ be a sequence of measurable functions on $latex [0,1]$ such that $latex |f_k(x)| < \infty$ a.e. x. Then there exists a sequence $latex c_k>0$ such that

$latex \dfrac{f_k(x)}{c_k} \to 0$ a.e. x.

Problem 5

Let $latex f(x,y)$ be separately continuous: continuous in each variable when the other one is fixed. Then f is measurable on $latex \R^2$.