A Kingman-like Theorem

TL;DR

This paper extends classical ergodic theorems to arbitrary measures and bounded observables, demonstrating the existence of Birkhoff's limit in specific continuous cases, with applications to Bowen's example.

Contribution

It introduces a Kingman-like theorem for finite measures and a Birkhoff's theorem variant for bounded observables, broadening ergodic theory applicability.

Findings

Established a Kingman-like theorem for arbitrary finite measures.

Proved Birkhoff's limit exists for certain continuous observables.

Applied results to Bowen's example demonstrating practical relevance.

Abstract

We provide a Kingman-like Theorem for arbitrary finite measures and a version of Birkhoff's Theorem for bounded observable. As an application, we show that Birkhoff's limit exists for some continuous observable, in an example of Bowen.