Homework 2, Real Analysis 2

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.

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