Finitely subadditive outer measures, finitely superadditive inner measures and their measurable sets

P D Stratigos

doi:10.1155/s016117129600066x

Abstract

Consider any set X. A finitely subadditive outer measure on P(X) is defined to be a function v from P(X) to R such that v(ϕ) = 0 and v is increasing and finitely subadditive. A finitely superadditive inner measure on P(X) is defined to be a function p from P(X) to R such that p(ϕ) = 0 and p is increasing and finitely superadditive (for disjoint unions) (It is to be noted that every finitely superadditive inner measure on P(X) is countably superadditive).This paper contributes to the study of finitely subadditive outer measures on P(X) and finitely superadditive inner measures on P(X) and their measurable sets.

Highlights

This paper contributes to the study of finitely subadditive outer measures on P(X) and finitely superadditive inner measures on 7:’(X) and their measurable sets
These considerations lead us to a general analysis of finitely subadditive outer measures on P(X) and finitely superadditive inner measures on 7(X) [For a finite outer measure on ’(X), it is wellknown that the class of u-measurable sets denoted by $ is a or-algebra, that v[s is a measure, and that for every set E, E E,_% iffu(E) uo(E) where u and vo are the outer measure and inner measure induced by vls,S0,S So The situation is vastly different in general for finitely subadditive outer measures, as we shall show
(1) Every regular outer measure v on P(X) has the following property For every sequence of sets (E./. if (E,) s increasing, lim, v(E, v(lim, E, (2) Every finite regular outer measure v on 7(X) has the following property For every set E, E 6,."; ifv(E) + u(E’) v(X) (,). We investigate these properties for finitely subadditive outer measures on T’(X) The first property is false in general The second property remains true even though there are two possibilities for defining regularity of v In the course of our investigations, we construct families of finitely subadditive outer measures on 7(X) which are not necessarily regular but which still have Property (,)

Summary

Introduction

This paper contributes to the study of finitely subadditive outer measures on P(X) and finitely superadditive inner measures on 7:’(X) and their measurable sets. We note that if is not normal, need not be a finitely subadditive outer measure it has been shown that for every element of M(), #, there exists an element of MR(E), ’, such that # < on Z: and #(X) ,(X), in general is not unique Different type proofs of this fact have been given in 11,7,6].

Results

Conclusion