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

TL;DR

This paper presents an elementary proof that finite continuous non-decreasing submodular functions on measurable spaces can be represented as a supremum of measures, using a totally ordered generating class of sets.

Contribution

It provides a new, elementary proof of the representation of submodular functions as supremums of measures, emphasizing the construction via standard measure extension theorems.

Findings

Measures attaining the supremum are unique.

The proof simplifies understanding of submodular function representations.

Polish spaces exemplify spaces with totally ordered generating classes.

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.