O-segments on topological measure spaces
Abstract
Description
Let $X$ be a topological space and $μ$ be a nonatomic finite measure on a $σ$-algebra $Σ$ containing the Borel $σ$-algebra of $X$. We say $μ$ is weakly outer regular, if for every $A \in Σ$ and $ε>0$, there exists an open set $O$ such that $μ(A \backslash O)=0$ and $μ(O \backslash A)<ε$. The main result of this paper is to show that if $f,g \in L^1(X,Σ, μ)$ with $\int_X f dμ=\int_X g dμ=1$, then there exists an increasing family of open sets $u(t)$, $t\in [0,1]$, such that $u(0)=\emptyset$, $u(1)=X$, and $\int_{u(t)} f dμ=\int_{u(t)} g dμ=t$ for all $t\in [0,1]$. We also study a similar problem for a finite collection of integrable functions on general finite and $σ$-finite nonatomic measure spaces.
10 pages
10 pages