A theory of integration for Ces\`aro limits

TL;DR

This paper develops a theoretical framework for spaces of sequences with Cesàro limits, linking them to integral operators and measure theory, and characterizes their structure and separability.

Contribution

It introduces and analyzes new spaces of sequences with Cesàro limits, establishing their isometric isomorphism to function spaces and exploring their measure-theoretic properties.

Findings

Spaces of sequences with Cesàro limits are characterized and linked to integral operators.

Isometric isomorphisms between sequence spaces and function spaces are established.

Conditions for separability of these spaces are identified.

Abstract

The Ces\`aro limit - the asymptotic average of a sequence of real numbers - is an operator of fundamental importance in probability, statistics and analysis. Surprisingly, spaces of sequences with Ces\`aro limits have not previously been studied. This paper introduces spaces of such sequences, denoted $K_p(\mathcal{A})$, with the Ces\`aro limit acting as a kind of integral. The space $\mathcal{F}$ comprised of all binary sequences with a Ces\`aro limit is studied first, along with the associated functional $\nu: \mathcal{F} \rightarrow [0,1]$ mapping each such sequence to its Ces\`aro limit. It is shown that $\mathcal{F}$ can be factored to produce a monotone class on which $\nu$ induces a countably additive set function. The space $K_p(\mathcal{A})$ is then defined, and a quotient denoted $\mathcal{K}_p(\mathcal{A})$ is shown to be isometrically isomorphic, under certain conditions, to the function space $\mathcal{L}_p(\mathbb{N},\mathcal{A},\nu)$, where $\mathcal{A}$ is a field of sets isomorphic to a subset of $\mathcal{F}$, and $\nu$ is a finitely additive measure induced by the functional mentioned above. The Ces\`aro limit of an element of $K_p(\mathcal{A})$ is shown to be equal to its integral. The complete $\mathcal{L}_p(\mathbb{N},\mathcal{A},\nu)$ spaces (and by implication, the $\mathcal{K}_p(\mathcal{A})$ spaces isomorphic to them) are characterised, and a sufficient condition for these spaces to be separable is identified.