# Selected topics in mathematical logic winter semester 2016/2017

Time: Tue, Thu 2:00pm–3:00pm

starts 2016‑10‑04

Place: KGRC

This course will serve as an introduction to measure-theoretic techniques in the study of countable Borel equivalence relations. The lectures will be largely self-contained, in particular employing only a minimal amount of descriptive set theory and avoiding the need for functional analysis altogether. We will meet from 2pm-3pm on Tuesdays and Thursdays, and the corresponding notes will be posted online.

The first month or so will be a crash course in measure theory, both on abstract families of sets (e.g., Caratheodory’s extension theorem, Fubini’s theorem, and the Radon-Nikodym theorem) and on Polish spaces (e.g., Lebesgue’s density theorem for Polish ultrametric spaces, the Polish space of Borel probability measures on a Polish space, and the ergodic decomposition of a Borel probability measure with respect to an equivalence relation).

The remainder of the course will concern basic ideas in the descriptive set-theoretic study of countable Borel equivalence relations (e.g., natural generalizations of Rokhlin’s lemma, graph colorings yielding the existence of maximal Borel sets with desirable properties, hyperfiniteness), invariant measures (e.g., the Glimm-Effros-style characterization of the existence of non-trivial sigma-finite Borel measures invariant with respect to a Borel cocycle, the Hopf-style characterization of the existence of Borel probability measures invariant with respect to a Borel cocycle, and the uniform ergodic decomposition of all Borel probability measures invariant with respect to a Borel cocycle), and measure hyperfiniteness.