A multiplicative measure on the positive real axis

2021/06/05 by Pablo Rocha, Rocha, Pablo
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #math.CA

paper · pdf · doi:10.48550/arxiv.2106.02784

11 pages

openalex publication_date 2021/06/05 · arxiv created 2021/06/15 · arxiv updated 2021/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we construct a measure μ on a σ-algebra M of subsets of the positive real axis, ℝ_>0, with the following multiplicative property: μ( \bigcup_j E_j ) = ∏_j μ(E_j) for every countable collection \ E_j \ of pairwise disjoint sets of M. For them, we apply the Carathéodory's procedure to the triplet ( ℝ_>0, ⋅ , τ), where ⋅ is the product of R and τ is the usual topology on ℝ_>0. We conclude this note describing the connection between this multiplicative measure μ and the Lebesgue measure.

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