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arXiv · 2406.18174

An elementary proof of representation of submodular function as an supremum of measures on $\sigma$-algebra with totally ordered generating class

Tetsuya Hattori
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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 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.

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BibTeXRIS

Tetsuya Hattori. 2024-06-26. An elementary proof of representation of submodular function as an supremum of measures on $\sigma$-algebra with totally ordered generating class. https://arxiv.org/abs/2406.18174

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