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 $X=[0,1]$, $\mathscr M = \mathscr B$, m 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.