Ultralimits of Birkhoff averages

TL;DR

This paper develops a dominated convergence theorem for ultralimits of integrable functions on compact metric spaces and applies it to establish a non-standard ergodic-like theorem for arbitrary probability measures.

Contribution

It introduces a new dominated convergence theorem for ultralimits and extends ergodic theorems to a non-standard setting for any probability measure.

Findings

Established a dominated convergence theorem for ultralimits.

Derived a non-standard ergodic-like theorem applicable to all probability measures.

Extended classical ergodic theory using ultralimit techniques.

Abstract

Given a compact metric space $X$ and a probability measure in the $\sigma-$algebra of Borel subsets of $X$, we will establish a dominated convergence theorem for ultralimits of sequences of integrable maps and apply it to deduce a non-standard ergodic-like theorem for any probability measure.