arXiv subjects

Jonathan M. Keith

Publications and source records attributed to Jonathan M. Keith.

8 recordsLinked to original sources

arXiv2026

On maximal families of independent sets with respect to asymptotic density

We study families of subsets of $\omega$ which are independent with respect to the asymptotic density $\mathsf{d}$. We show, for instance, that there exists a maximal $\mathsf{d}$-independent family $\mathcal{A}$ such that $\mathsf{d}[\mathcal{A}]$ attains a prescribed set of values in $(0,1)$ with at most countably many exceptions. In addition, under $\mathrm{cov}(\mathcal{N})=\mathfrak{c}$, it is possible to construct such $\mathcal{A}$ with no exceptions. We also construct $2^{\mathfrak{c}}$ maximal $\mathsf{d}$-independent families with pairwise distinct generated density fields and obtain maximal families with strong definability pathologies, including examples without the Baire property and, consistently, nonmeasurable examples.

math.LO↗

arXiv2025

Completeness and additive property for submeasures

Given an extended real-valued submeasure $\nu$ defined on a field of subsets $\Sigma$ of a given set, we provide necessary and sufficient conditions for which the pseudometric $d_\nu$ defined by $d_{\nu}(A,B):=\min\{1,\nu(A\bigtriangleup B)\}$ for all $A,B \in \Sigma$ is complete. As an application, we show that if $\varphi: \mathcal{P}(\omega)\to [0,\infty]$ is a lower semicontinuous submeasure and $\nu(A):=\lim_n \varphi(A\setminus \{0, 1, \ldots, n-1\})$ for all $A\subseteq \omega$, then $d_\nu$ is complete. This includes the case of all weighted upper densities, fixing a gap in a proof by Just and Krawczyk in [Trans.~Amer.~Math.~Soc.~\textbf{285} (1984), 803--816]. In contrast, we prove that if $\nu$ is the upper Banach density (or an upper density greater than or equal to the latter) then $d_\nu$ is not complete. We conclude with several characterizations of completeness in terms of the Stone space of the Boolean algebra $\Sigma/\nu$.

math.FA↗

arXiv2024

The pseudometric topology induced by upper asymptotic density

Upper asymptotic density induces a pseudometric on the power set of the natural numbers, with respect to which $P(\mathbb{N})$ is complete. The collection $D$ of sets with asymptotic density is closed in this pseudometric, and closed subsets of $D$ are characterised by a generalisation of an additivity property (AP0).

math.GN↗

arXiv2023

On the measurability of a numerical function with respect to a family of sets

The following document is a translation (from French to English) of: Gabriele H. Greco, Sur la mesurabilit\'e d'une fonction num\'erique par rapport \`a une famille d'ensembles, Rendiconti del Seminario Matematico della Universit\`a di Padova}, tome 65 (1981), pp. 163--176. Translated by: Jonathan M. Keith, School of Mathematics, Monash University, jonathan.keith@monash.edu. With thanks to: Prof. Andrea D'Agnolo, Editor-in-Chief of the above journal, for permission to publish this translation.

math.FA↗

arXiv2023

Measurable functions on charge spaces

The concept of measurability of functions on a charge space is generalised for functions taking values in a uniform space. Several existing forms of measurability generalise naturally in this context, and new forms of measurability are proposed. Conditions under which the various forms of measurability are logically equivalent are identified. Applying these concepts to real-valued functions, some recent characterisations of measurable functions on a bounded charge space are generalised to the unbounded case.

math.FA↗

arXiv2022

A theory of integration for Cesàro limits

The Cesàro 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 Cesàro limits have not previously been studied. This paper introduces spaces of such sequences, denoted $K_p(\mathcal{A})$, with the Cesàro limit acting as a kind of integral. The space $\mathcal{F}$ comprised of all binary sequences with a Cesàro limit is studied first, along with the associated functional $ν: \mathcal{F} \rightarrow [0,1]$ mapping each such sequence to its Cesàro limit. It is shown that $\mathcal{F}$ can be factored to produce a monotone class on which $ν$ 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 $\mathcal{L}_p(\mathbb{N},\mathcal{A},ν)$, where $\mathcal{A}$ is a field of sets isomorphic to a subset of $\mathcal{F}$, and $ν$ is a finitely additive measure induced by the functional mentioned above. The Cesàro limit of an element of $K_p(\mathcal{A})$ is shown to be equal to its integral. The complete $\mathcal{L}_p(\mathbb{N},\mathcal{A},ν)$ 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.

math.CA↗

arXiv2021

Properties of functions on a bounded charge space

A charge space $(X,\mathcal{A},μ)$ is a generalisation of a measure space, consisting of a sample space $X$, a field of subsets $\mathcal{A}$ and a finitely additive measure $μ$, also known as a charge. Key properties a real-valued function on $X$ may possess include $T_1$-measurability and integrability. These properties are generalisations of corresponding properties of real-valued functions on a (countably additive) measure space. However, these properties are less well studied than their measure-theoretic counterparts. This paper describes new characterisations of $T_1$-measurability and integrability in the case that the charge space is bounded, that is, $μ(X) < \infty$. These characterisations are convenient for analytic purposes; for example, they facilitate simple proofs that $T_1$-measurability is equivalent to conventional measurability and integrability is equivalent to Lebesgue integrability, if $(X,\mathcal{A},μ)$ is a complete measure space. Several additional contributions to the theory of bounded charges are also presented. New characterisations of equality almost everywhere of two real-valued functions on a bounded charge space are provided. Necessary and sufficient conditions for the function space $L_1(X,\mathcal{A},μ)$ to be a Banach space are determined. Lastly, the concept of completion of a measure space is generalised for charge spaces, and it is shown that under certain conditions, completion of a charge space adds no new equivalence classes to the quotient space $\mathcal{L}_p(X,\mathcal{A},μ)$.

math.FA↗

arXiv2021

Binary sequences with a Cesàro limit

The Cesàro 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 Cesàro limits, this paper considers the space $\mathcal{F}$ comprised of all binary sequences with a Cesàro limit, and the associated functional $ν: \mathcal{F} \rightarrow [0,1]$ mapping each such sequence to its Cesàro limit. The basic properties of $\mathcal{F}$ and $ν$ are enumerated, and chains (totally ordered sets) in $\mathcal{F}$ on which $ν$ is countably additive are studied in detail. The main result of the paper concerns a structural property of the pair $(\mathcal{F},ν)$, specifically that $\mathcal{F}$ can be factored (in a certain sense) to produce a monotone class on which $ν$ is countably additive. In the process, a slight generalisation and clarification of the monotone class theorem for Boolean algebras is proved.

math.FA↗