Pointwise ergodic theorem for locally countable quasi-pmp graphs

Anush Tserunyan

Published 2018-05-07Version 1

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, analogous to pointwise ergodic theorems for group actions, replacing the group with a Schreier graph of the action. Our theorem equates the global condition of ergodicity to locally approximating the means of $L^1$-functions via an increasing sequence of Borel subgraphs with finite connected components. Equivalently, it states that any (not necessarily pmp) locally countable Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. The pmp version of this theorem was first proven by R. Tucker-Drob using probabilistic methods. Our proof is different: it is descriptive set theoretic and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant, a method of producing finite equivalence subrelations with large domain, and a simple method of exploiting nonamenability of a measured graph. The non-pmp setting additionally requires a new gadget for analyzing the interplay between the underlying cocycle and the graph.