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Homework 12, Real Analysis 2

Due May 18

Problem 1

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

  1. Let $latex Q_r$ be a translate of the cube $latex \{x\in\R^d: 0 < x_j \le r, j=1,\ldots,d\}.$ If $latex \mu(Q_1) = c$, then $latex \mu(Q_{1/n}) = c/n^d$ for each integer n.

  2. $latex \mu$ is absolutely continuous with respect to m, and there is a locally integrable function f such that $latex \displaystyle \mu(E) = \int_E f dx.$

  3. By the differentiation theorem, is follows that $latex f(x) = c$ a.e., and hence $latex \mu = cm$.

Problem 2

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

  1. If $latex \nu_1\perp\mu$ and $latex \nu_2\perp\mu$, then $latex \nu_1+\nu_2 \perp \mu$.

  2. If $latex \nu_1\ll \mu$ and $latex \nu_2\ll\mu$, then $latex \nu_1 + \nu_2 \ll\mu$.

  3. If $latex \nu_1\perp\nu_2$ then $latex |\nu_1|\perp|\nu_2|$.

  4. $latex \nu\ll|\nu|$

  5. If $latex \nu\perp\mu$ and $latex \nu\ll\mu$, then $latex \nu=0$.

Problem 3

Let $latex X=[0,1]$, $latex \mathscr M = \mathscr B$, m Lebesgue measure and $latex \mu$ counting measure on $latex \mathscr M$.

  1. $latex m\ll\mu$ but $latex dm\not=f d\mu$ for any f.

  2. $latex \mu$ has no decomposition with respect to m.

Problem 4

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

  1. Any finite subset of $latex L^1(\mu)$ is uniformly integrable.

  2. If $latex (f_n)$ is a convergente sequence in $latex L^1(\mu)$, then it is uniformly integrable.