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.

Post Tagged with 

Comments & Responses

Leave a Reply