A multiplicative measure on the positive real axis

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: \[ μ\left( \bigcup_j E_j \right) = \prod_j μ(E_j) \] for every countable collection $\{E_j\}$ of pairwise disjoint sets of $\mathcal{M}$. For them, we apply the Carathéodory'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.

11 pages

References in corpus (1)

• Novel Approach to Infinite Products Using Multiplicative Modulus Function