# Homework 13, Real Analysis 2

## Due May 25

### Problem 1

Suppose $\tau$ is measure-preserving, with $\mu(X) = 1$. If E is invariant, then there exists a set E’ so that $E' = \tau^{-1}(E')$, and E and E’ differ by a set of measure zero.

### Problem 2

Let $\tau$ be measure-preserving, with $\mu(X)=1$. Then $\tau$ is ergodic if and only if whenever $\nu$ is absolutely continuous with respect to $\mu$ and $\nu$ is invariant (that is $\nu(\tau^{-1}(E) = \nu(E)$ for all measurable E), then $\nu = c\mu$), then $\nu = c\mu$ for some constant c.

### Problem 3

The Hausdorff measure $\mathscr H^\alpha$ is not $\sigma$-finite on $\R^d$ if $\alpha < d$.

### Problem 4

Let $\{E_k\}$ be a sequence of Borel sets in $\R^d$. If $\dim E_k\le\alpha$ for all k, then

$\displaystyle \dim \bigcup E_k \le \alpha$.

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