Tarski Measure

TL;DR

This paper develops a category-theoretic framework for stationary measures, embedding sizes into an algebraic structure and providing a new algebraic characterization of classical measures, thus extending Tarski's foundational work.

Contribution

It introduces a novel algebraic and categorical approach to stationary measures, unifying measure theory with inverse monoid structures and extending Tarski's foundational ideas.

Findings

Sizes form a partially ordered commutative inverse monoid

Basic measure theory theorems are proved in this framework

Classical real-valued stationary measures are characterized algebraically

Abstract

We present an approach to the study of stationary measures placing Tarski's foundational work in this area within a modern category theoretic context. Guiding this work is the notion that measurable spaces equipped with symmetries carry an intrinsic notion of `size'. We demonstrate that within such measurable spaces, the collection of these sizes embeds into a partially ordered commutative inverse monoid. Using this we prove, in great generality, the basic theorems of measure theory and give an algebraic characterization of the space of classical real valued stationary measures.