Pseudoquotient extensions of measure spaces

TL;DR

This paper introduces pseudoquotient spaces built from measure spaces and investigates conditions under which the measure can be extended from the original space to these pseudoquotients, broadening the scope of measure theory.

Contribution

It defines pseudoquotient spaces for measure spaces and explores the extension of measures to these new structures, a novel approach in measure theory.

Findings

Conditions for measure extension are identified.

Pseudoquotient spaces generalize measure spaces.

Framework for measure extension is established.

Abstract

A space of pseudoquotients $\mathcal P (X,S)$ is defined as equivalence classes of pairs $(x,f)$, where $x$ is an element of a non-empty set $X$, $f$ is an element of $S$, a commutative semigroup of injective maps from $X$ to $X$, and $(x,f) \sim (y,g)$ if $gx=fy$. In this note we assume that $(X,\Sigma,\mu)$ is a measure space and that $S$ is a commutative semigroup of measurable injections acting on $X$ and investigate under what conditions there is an extension of $\mu$ to $\mathcal P (X,S)$.