On finitely subadditive outer measures and modularity properties

Charles Traina

doi:10.1155/s0161171203208115

Abstract

Let ν be a finite, finitely subadditive outer measure on P(X). Define ρ (E) = ν (X) − ν (E′) for E ⊂ X. The measurable sets Sν and Sρ and the set S = {E ⊂ X/ν (E) = ρ (E)} are investigated in general, and in the presence of regularity or modularity assumptions on ν. This is also done for ν0(E) = inf{ν (M)/E ⊂ M ∈ Sν }. General properties of ν are derived when ν is weakly submodular. Applications and numerous examples are given.

Highlights

Let X be an arbitrary nonempty set and ν a finite, finitely subadditive outer measure on P (X). ρ denotes the set function defined by ρ(E) = ν(X) − ν(E ), E ⊂ X
Throughout this section and the rest of the paper, ν will designate a finite-valued, finitely subadditive outer measure defined on the power set P (X) of a nonempty set X. ρ will designate the associated set function ρ(E) = ν(X) − ν(E ), where E ⊂ X, and the prime will designate complement
In order to see when ρ is an inner measure, we introduce the following definition

Summary

Introduction

Let ν be a finite, finitely subadditive outer measure, ν is regular if, for any E ⊂ X, there exists an M ∈ Sν such that E ⊂ M and ν(E) = ν(M). We can say that any regular outer measure satisfies condition (2.3). For E ⊂ X, is a finite and finitely subadditive outer measure on P (X) and μ is submodular on P (X).

Results

Conclusion