O-segments on topological measure spaces

TL;DR

This paper proves the existence of a continuous family of open sets in a measure space that interpolates between two integrable functions with equal total integral, extending to collections of functions.

Contribution

It introduces a method to construct increasing open set families matching integrals of functions, generalizing previous measure-theoretic interpolation results.

Findings

Existence of open set families interpolating between functions

Extension to collections of functions on nonatomic measure spaces

Applicable to finite and sigma-finite 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 $\int_X f d\mu=\int_X g d\mu=1$, then there exists an increasing family of open sets $u(t)$, $t\in [0,1]$, such that $u(0)=\emptyset$, $u(1)=X$, and $\int_{u(t)} f d\mu=\int_{u(t)} g d\mu=t$ 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.