Homework 12, Real Analysis 2

Due May 18

Problem 1

The purpose of the following exercises is to prove the following statement: If \mu is a translation-invariant Borel measure on \R^d that is finite on compact sets, then \mu is a multiple of Lebesgue measure.

  1. Let Q_r be a translate of the cube \{x\in\R^d: 0 < x_j \le r, j=1,\ldots,d\}. If \mu(Q_1) = c, then \mu(Q_{1/n}) = c/n^d for each integer n.
  2. \mu is absolutely continuous with respect to m, and there is a locally integrable function f such that \displaystyle \mu(E) = \int_E f dx.
  3. By the differentiation theorem, is follows that f(x) = c a.e., and hence \mu = cm.

Problem 2

Suppose \nu, \nu_1, \nu_2 are signed measures on (X,\mathscr M) and \mu a positive measure.

  1. If \nu_1\perp\mu and \nu_2\perp\mu, then \nu_1+\nu_2 \perp \mu.
  2. If \nu_1\ll \mu and \nu_2\ll\mu, then \nu_1 + \nu_2 \ll\mu.
  3. If \nu_1\perp\nu_2 then |\nu_1|\perp|\nu_2|.
  4. \nu\ll|\nu|
  5. If \nu\perp\mu and \nu\ll\mu, then \nu=0.

Problem 3

Let X=[0,1], \mathscr M = \mathscr Bm Lebesgue measure and \mu counting measure on \mathscr M.

  1. m\ll\mu but dm\not=f d\mu for any f.
  2. \mu has no decomposition with respect to m.

Problem 4

Let \mu be a positive measure. A collection of integrable functions \{f_\alpha\} is called uniformly integrable if for every \e>0 there exists \delta>0 such that |\int_E f_\alpha d\mu|<\e for all \alpha whenever \mu(E)<\delta.

  1. Any finite subset of L^1(\mu) is uniformly integrable.
  2. If (f_n) is a convergente sequence in L^1(\mu), then it is uniformly integrable.
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