O-segments on topological measure spaces
Mohammad Javaheri

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 be a topological space and be a nonatomic finite measure on a -algebra containing the Borel -algebra of . We say is weakly outer regular, if for every and , there exists an open set such that and . The main result of this paper is to show that if with , then there exists an increasing family of open sets , , such that , , and for all . We also study a similar problem for a finite collection of integrable functions on general finite and -finite nonatomic measure spaces.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Advanced Operator Algebra Research
