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A density version of a theorem of Banach

David A. Ross

TL;DR

This paper develops a density-variant of Banach’s representation theorem using nonstandard analysis. It introduces the $d$-limit concept with upper density $\bar{d}$ and proves the equivalence between weak-$d$ convergence of a uniformly bounded sequence of functions and a density-limited double-limit, via a density-enhanced Bergelson lemma and Loeb/S-measures. The approach transfers finite-additive, nonstandard information to a density framework, yielding a density version of Banach’s theorem and its representation form. The results provide a robust tool for analyzing density-based convergence phenomena with potential applications in ergodic theory and functional analysis.

Abstract

The S-measure construction from nonstandard analysis is used to prove an extension of a result on the intersection of sets in a finitely-additive measure space. This is then used to give a density-limit version of a representation theorem of Banach.

A density version of a theorem of Banach

TL;DR

This paper develops a density-variant of Banach’s representation theorem using nonstandard analysis. It introduces the -limit concept with upper density and proves the equivalence between weak- convergence of a uniformly bounded sequence of functions and a density-limited double-limit, via a density-enhanced Bergelson lemma and Loeb/S-measures. The approach transfers finite-additive, nonstandard information to a density framework, yielding a density version of Banach’s theorem and its representation form. The results provide a robust tool for analyzing density-based convergence phenomena with potential applications in ergodic theory and functional analysis.

Abstract

The S-measure construction from nonstandard analysis is used to prove an extension of a result on the intersection of sets in a finitely-additive measure space. This is then used to give a density-limit version of a representation theorem of Banach.

Paper Structure

This paper contains 5 sections, 8 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Loeb measures and S--measures
  3. A fundamental lemma
  4. Proof of Lemma \ref{['weakbergelson']}
  5. Proof of Theorem \ref{['thm4']}

Key Result

Theorem 1.1

Let $X$ be a set, $B(X)$ be all bounded real functions on $X$, and $\{\,f_n : n\in\mathbb{N}\}$ be a uniformly bounded sequence. The following are equivalent: (i) $\{\,f_n\}_n$ converges weakly to $0$; (ii) for any sequence $\{x_k : k\in\mathbb{N}\}$ in $X$, $\lim\limits_{n\to\infty}\liminf\limits_{

Theorems & Definitions (10)

  • Theorem 1.1
  • Corollary 1
  • Theorem 1.2
  • Corollary 2
  • Lemma 1
  • Lemma 2
  • Corollary 3
  • proof
  • Corollary 4
  • proof