The pointwise ergodic theorem on finitely additive spaces
Morenikeji Neri
TL;DR
This paper extends Birkhoff's pointwise ergodic theorem to finitely additive probability spaces by introducing finite almost sure convergence, a notion tailored to finitely additive measures. It proves that if ergodic averages $A_n f$ satisfy a natural measurability condition, then $\{A_n f\}$ converges in the finitely almost sure sense, using a finite Calderón transference principle and quantitative oscillation/mermetastability analysis. The work delivers a quantitative framework, including a learnable rate of uniform convergence $\delta(\lambda,\varepsilon)$ and a metastability bound $\Phi(\lambda,\varepsilon,g)$, and connects these results to proof mining; it also shows how classical σ-additive techniques can be adapted to the finitely additive setting. Altogether, the results broaden ergodic theory to finitely additive spaces and provide explicit quantitative bounds, with potential implications for logical analysis of stochastic limit theorems.
Abstract
The almost sure convergence of ergodic averages in Birkhoff's pointwise ergodic theorem is known to fail in the finitely additive setting. We introduce a natural reformulation of almost sure convergence suitable for finitely additive measures, which we call finite almost sure convergence. Unlike the classical formulation, finite almost sure convergence only involves measures of finite unions and intersections, making it well adapted to finitely additive spaces. Using this notion, we extend the pointwise ergodic theorem to finitely additive probability spaces. Our proof relies on demonstrating that several quantitative generalizations of the pointwise ergodic theorem remain valid in the finitely additive setting via an extension of the Calderón transference principle. The result then follows by exploiting the relationships between quantitative notions of almost sure convergence developed by the author and Powell (c.f. Trans. Amer. Math. Soc. Series B 12 (2025), 974-1019).