2024/06/26 by Tetsuya Hattori, Hattori, Tetsuya
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Economics and business #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Probability (math.PR) #Risk Management (q-fin.RM)
paper · pdf · doi:10.48550/arxiv.2406.18174
openalex publication_date 2024/06/26 · openalex created_date 2024/06/28 · openalex updated_date 2026/07/28
We give an alternative proof of a fact that a finite continuous non-decreasing submodular set function on a measurable space can be expressed as a supremum of measures dominated by the function, if there exists a class of sets which is totally ordered with respect to inclusion and generates the sigma-algebra of the space. The proof is elementary in the sense that the measure attaining the supremum in the claim is constructed by a standard extension theorem of measures. As a consequence, a uniquness of the supremum attaining measure also follows. A Polish space is an examples of the measurable space which has a class of totally ordered sets that generates the Borel sigma-algebra.