Ir al contenido principal

Due February 16

Problem 1

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

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

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

Problem 2

Let C be the Cantor set.

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

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

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

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

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

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

Problem 4

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

$\dfrac{f_k(x)}{c_k} \to 0$ a.e. x.

Problem 5

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