Abstract
Dating back to Birkhoff, pointwise ergodic theorems for probability measure preserving (pmp) actions of countable groups are bridges between the global condition of ergodicity (measure-transitivity) and the local combinatorics of the actions. Each such action induces a Borel equivalence relation with countable classes and the study of these equivalence relations is a flourishing subject in modern descriptive set theory. Such an equivalence relation can also be viewed as the connectedness relation of a locally countable Borel graph. These strong connections between equivalence relations, group actions, and graphs create an extremely fruitful interplay between descriptive set theory, ergodic theory, measured group theory, probability theory, and descriptive graph combinatorics. I will discuss how descriptive set theoretic thinking combined with combinatorial and measure theoretic arguments yields a pointwise ergodic theorem for quasi-pmp locally countable graphs. This theorem is a vastly general random version of pointwise ergodic theorems for group actions and is provably the best possible pointwise ergodic result for some of these actions.