Due May 25
Suppose is measure-preserving, with . If E is invariant, then there exists a set E’ so that , and E and E’ differ by a set of measure zero.
Let be measure-preserving, with . Then is ergodic if and only if whenever is absolutely continuous with respect to and is invariant (that is for all measurable E), then ), then for some constant c.
The Hausdorff measure is not -finite on if .
Let be a sequence of Borel sets in . If for all k, then