The pointwise ergodic theorem on finitely additive spaces

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 Calder\'on 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).