On the Koopman-von Neumann convergence condition of Ces\`{a}ro means in Riesz spaces

TL;DR

This paper extends the Koopman-von Neumann convergence condition for Cesàro means to Dedekind complete Riesz spaces, providing new insights into weak mixing and convergence in L^1.

Contribution

It generalizes the convergence condition to Riesz spaces and characterizes conditional weak mixing within this framework.

Findings

Extended convergence condition to Dedekind complete Riesz spaces.

Provided a characterization of conditional weak mixing in Riesz spaces.

Applied results to convergence in L^1.

Abstract

We extend the Koopman-von Neumann convergence condition on the Ces\`{a}ro mean to the context of a Dedekind complete Riesz space with weak order unit. As a consequence, a characterisation of conditional weak mixing is given in the Riesz space setting. The results are applied to convergence in $L^1$.