Embedding measure spaces

TL;DR

This paper develops a method to construct all measure spaces into which a given measure space can be embedded, inspired by the ultrafilter construction of the Stone–Čech compactification, with simplifications under certain conditions.

Contribution

It introduces a novel construction of measure spaces embedding a given measure space, extending the ultrafilter approach from topology to measure theory.

Findings

Construction of embedding measure spaces detailed

Simplifications occur under specific conditions

Examples illustrating the construction provided

Abstract

For a given measure space $(X,{\mathscr B},\mu)$ we construct all measure spaces $(Y,{\mathscr C},\lambda)$ in which $(X,{\mathscr B},\mu)$ is embeddable. The construction is modeled on the ultrafilter construction of the Stone--\v{C}ech compactification of a completely regular topological space. Under certain conditions the construction simplifies. Examples are given when this simplification occurs.