A Carath\'eodory's Extension Theorem for Families Simpler Than an Algebras of Sets

TL;DR

This paper introduces a new extension theorem for probability measures that simplifies the process by using a refinement structure instead of an algebra, easing the definition of measures over complex set families.

Contribution

It develops an extension theorem similar to Carathéodory's but applicable to quasi-measures over refinements, simplifying measure construction.

Findings

The new theorem allows measure extension from simpler refinements.

It reduces complexity in defining measures over targeted set families.

The approach facilitates easier measure construction in probability theory.

Abstract

The Carath\'eodory's Extension Theorem is a powerful tool that allows us to generate a measure, over a sigma-algebra, from a pre-measure defined over an algebra of sets. However, although this result reduces our work to define a measure by only needing to define a pre-measure, it is not always easy to define the latter. The problem occurs when taking the smallest algebra that contains a family of targeted sets, it can be very complicated to consistently define the value of the pre-measure over its finite union of these sets - a union that is an element of the algebra. Thus, our objective in this article is to reproduce an extension theorem, just like the Carath\'eodory's Extension Theorem, but in the context of probability measures and replacing the need for a probability pre-measure defined over an algebra for now a quasi-measure defined over a refinement. The gain, then, is that the \textit{manual elaboration} of a quasi-measure is simpler than the elaboration of a pre-measure, since a refinement is a simpler structure than an algebra.