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Homework 13, Real Analysis 2

Due May 25

Problem 1

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

Problem 2

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

Problem 3

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

Problem 4

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

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