O-segments on topological measure spaces

Abstract

Let $X$ be a topological space and $\mu$ be a nonatomic finite measure on a $\sigma$-algebra $\Sigma$ containing the Borel $\sigma$-algebra of $X$. We say $\mu$ is weakly outer regular, if for every $A\in \Sigma$ and $\epsilon >0$, there exists an open set $O$ such that $\mu (A\backslash O)=0$ and $\mu (O\backslash A)<\epsilon$. The main result of this paper is to show that if $f,g\in L^1(X,\Sigma ,\mu )$ with ${\int }_{X}fd\mu ={\int }_{X}gd\mu =1$, then there exists an increasing family of open sets $u(t)$, $t\in [0,1]$, such that $u(0)=\varnothing$, $u(1)=X$, and ${\int }_{u(t)}fd\mu ={\int }_{u(t)}gd\mu =t$ for all $t\in [0,1]$. We also study a similar problem for a finite collection of integrable functions on general finite and $\sigma$-finite nonatomic measure spaces.

10 pages