A note on product of measures

TL;DR

This paper introduces a modified definition of the product of measures that is associative, applicable to all measures including non-$\sigma$-finite, and aligns with the Fubini--Tonelli theorem.

Contribution

It proposes a new, unique, and associative measure product definition that extends to arbitrary measures and maintains consistency with classical theorems.

Findings

Defines a modified product of measures that is associative.

Ensures the product applies to non-$\sigma$-finite measures.

Maintains compatibility with Fubini--Tonelli theorem.

Abstract

A slight modification to Halmos' definition of product of measures yields a uniquely characterized associative product. The operation applies to arbitrary (not necessarily $\sigma-$finite) measures and is consistent with the Fubini--Tonelli theorem.