Good measures and Fraisse theory

A full probability measure m on the Cantor space C is called good if there exists a uniquely ergodic homeomorphism of C whose unique invariant measure is m. Ethan Akin proved that every good measure is determined by its clopen-value set, i.e., the set of its values on clopen subsets of C.  He also showed that for so-called Q-like clopen-value sets, there is a generic measure preserving homeomorphism of C. We employ Fraisse theory to give short proofs of these results, and provide a full characterization of good measures with the clopen-value set contained in the rationals for which there exists a generic measure preserving homeomorphism. We also study finitely-additive measures on Boolean algebras with values in a given semigroup.

This is joint work in progress with Michal Doucha, Dominik Kwietniak and Piotr Niemiec.