A Nonseparable Invariant Extension of Lebesgue Measure -- A Generalized and Abstract Approach

TL;DR

This paper presents a generalized, abstract framework for extending Lebesgue measure to nonseparable spaces using combinatorial set theory techniques, expanding on classical theorems by Kakutani and Oxtoby.

Contribution

It introduces a novel abstract approach to nonseparable measure extension, utilizing combinatorial set theory and modified small set notions.

Findings

Established a generalized theorem for nonseparable Lebesgue measure extension

Developed methods involving independent families of sets and small set modifications

Extended classical measure extension results to broader, more abstract spaces

Abstract

Here using some methods of combinatorial set theory, particularly the ones related to the construction of independent families of sets and some modified version of the notion of small sets originally introduced by Riecan, Riecan and Neubrunn, we give abstract and generalized formulation of a remarkable theorem of Kakutani and Oxtoby relating to nonseparable extension of Lebesgue measure in spaces with transformation groups