A multiplicative measure on the positive real axis

TL;DR

This paper constructs a measure on the positive real axis with a multiplicative property for disjoint sets, using Carathéodory's procedure, and explores its relation to Lebesgue measure.

Contribution

It introduces a novel multiplicative measure on \\mathbb{R}_{>0} using Carathéodory's method, highlighting its connection to Lebesgue measure.

Findings

Defined a measure with multiplicative property on positive reals

Applied Carathéodory's construction to this measure

Explored the relationship between the new measure and Lebesgue measure

Abstract

In this note we construct a measure $\mu$ on a $\sigma$-algebra $\mathcal{M}$ of subsets of the positive real axis, $\mathbb{R}_{>0}$, with the following multiplicative property: \[ \mu \left( \bigcup_j E_j \right) = \prod_j \mu(E_j) \] for every countable collection $\{ E_j \}$ of pairwise disjoint sets of $\mathcal{M}$. For them, we apply the Carath\'eodory's procedure to the triplet $\left( \mathbb{R}_{>0}, \cdot \, , \tau \right)$, where $\cdot$ is the product of R and $\tau$ is the usual topology on $\mathbb{R}_{>0}$. We conclude this note describing the connection between this multiplicative measure $\mu$ and the Lebesgue measure.