An elementary proof of representation of submodular function as an supremum of measures on $σ$-algebra with totally ordered generating class
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911933598269440 |
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| author | Hattori, Tetsuya |
| author_facet | Hattori, Tetsuya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An elementary proof of representation of submodular function as an supremum of measures on $σ$-algebra with totally ordered generating class Hattori, Tetsuya Functional Analysis Probability Risk Management 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. |
| title | An elementary proof of representation of submodular function as an supremum of measures on $σ$-algebra with totally ordered generating class |
| topic | Functional Analysis Probability Risk Management |
| url | https://arxiv.org/abs/2406.18174 |