Paper: arXiv 2406.18174

Authors: Tetsuya Hattori

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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This is a purely theoretical measure-theoretic paper that provides an elementary alternative proof for the representation of submodular set functions using standard measure extension theorems instead of the Hahn-Banach Theorem. While mathematically rigorous and clearly written, it contains no empirical data, backtests, or financial modeling applications beyond brief theoretical references to Choquet integrals and coherent risk measures.

Research Flowchart

  flowchart TD
    A["<b>Research Goal</b><br/>Elementary proof for submodular set function<br/>represented as supremum of measures"] --> B["<b>Inputs & Assumptions</b><br/>Measurable space with totally ordered<br/>generating class & finite submodular function"]
    
    B --> C["<b>Construction Process</b><br/>Define a set function on the<br/>totally ordered generating class"]
    
    C --> D["<b>Measure Extension</b><br/>Apply standard Measure Extension Theorem<br/>to build the supremum-attaining measure"]
    
    D --> E["<b>Key Findings & Outcomes</b><br/>• Elementary proof established<br/>• Uniqueness of attaining measure proven<br/>• Validated for Polish spaces (Borel σ-algebra)"]

Cited References (5)

  1. Coherent Measures of Risk (1999) — link
  2. Cores of convex games (1971) — link
  3. Coherent Risk Measures on General Probability Spaces (2002) — link
  4. On law invariant coherent risk measures (2001) — link
  5. (cid:26) non-additive measure and integral (1994)

0 of 5 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.