Binary sequences with a Ces\`aro limit

TL;DR

This paper investigates the structure of binary sequences with a Cesàro limit, exploring their properties, chains, and a factorization that yields a monotone class where the Cesàro limit functional is countably additive.

Contribution

It introduces a detailed analysis of the space of binary sequences with Cesàro limits and proves a new factorization result related to monotone classes and countable additivity.

Findings

Characterization of the space of binary sequences with Cesàro limits

Identification of chains where the Cesàro limit functional is countably additive

A new factorization of the sequence space leading to a monotone class with countable additivity

Abstract

The Ces\`aro limit - the asymptotic average of a sequence of real numbers - is an operator of fundamental importance in probability, statistics and mathematical analysis. To better understand sequences with Ces\`aro limits, this paper considers the space $\mathcal{F}$ comprised of all binary sequences with a Ces\`aro limit, and the associated functional $\nu: \mathcal{F} \rightarrow [0,1]$ mapping each such sequence to its Ces\`aro limit. The basic properties of $\mathcal{F}$ and $\nu$ are enumerated, and chains (totally ordered sets) in $\mathcal{F}$ on which $\nu$ is countably additive are studied in detail. The main result of the paper concerns a structural property of the pair $(\mathcal{F},\nu)$, specifically that $\mathcal{F}$ can be factored (in a certain sense) to produce a monotone class on which $\nu$ is countably additive. In the process, a slight generalisation and clarification of the monotone class theorem for Boolean algebras is proved.