Elementary Proof of Representation of Submodular Function as Supremum of Measures on $\sigma$-Algebra with Totally Ordered Generating Class
Vol. 31 (2024), No. 2, Page 167–180.
Hattori, Tetsuya
Elementary Proof of Representation of Submodular Function as Supremum of Measures on $\sigma$-Algebra with Totally Ordered Generating Class
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Abstract:
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 chain (class of sets which is totally ordered with respect to inclusion) which 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, unique existence of the supremum attaining measure follows. A Polish space is an example of a measurable space which has a chain that generates the Borel sigma-algebra.
Keywords: submodular function, convex game, risk measure, measurable space.
Mathematics Subject Classification (2020): Primary 60A10; Secondary 60Axx.
Received: 2023-10-23
