A note on Carath\'eodory's Extension Theorem

TL;DR

This paper extends Carathéodory's extension theorem to a broader class of subsets called quasi-semi-rings, relaxing previous restrictions and potentially broadening its applicability in measure theory.

Contribution

It introduces the concept of quasi-semi-rings and proves that Carathéodory's extension theorem remains valid for this new class.

Findings

Carathéodory's theorem applies to quasi-semi-rings

Broader class of sets for measure extension

Potential applications in measure theory and probability

Abstract

In this note, we show that the Carath\'eodory's extension theorem is still valid for a class of subsets of $\Omega$ less restricted than a semi-ring, which we call quasi-semi-ring.