A Realization of Measurable Sets as Limit Points

TL;DR

This paper presents a novel approach to understanding measurable sets by representing them as limit points of Cauchy sequences within an algebra, using a pseudometric derived from a sigma finite measure.

Contribution

It introduces a new perspective on measurable sets as limit points, connecting measure theory with metric space concepts through a pseudometric framework.

Findings

Measurable sets can be characterized as limit points of Cauchy sequences.

A pseudometric based on a sigma finite measure facilitates this characterization.

The approach bridges measure theory and metric space analysis.

Abstract

Starting with a sigma finite measure on an algebra, we define a pseudometric and show how measurable sets from the Caratheodory Extension Theorem can be thought of as limit points of Cauchy sequences in the algebra.