Measurability of Intersections of Measurable Multifunctions

TL;DR

This paper establishes the universal compact-measurability of intersections of certain measurable multifunctions, introduces a maximal part of a multifunction, and applies these results to spectral theory of measurable operator families.

Contribution

It generalizes previous results on intersections of measurable multifunctions and introduces a new concept of the maximal part of a multifunction with measurability properties.

Findings

Proves universal compact-measurability of intersections of measurable multifunctions.

Introduces the maximal part of a multifunction on a quotient space.

Applies results to spectral theory of measurable families of operators.

Abstract

We prove universal compact-measurability of the intersection of a compact-measurable Souslin family of closed-valued multifunctions. This generalizes previous results on intersections of measurable multifunctions. We introduce the unique maximal part of a multifunction which is defined on the quotient given by an equivalence relation. Measurability of this part of a multifunction is proven in a special case. We show how these results apply to the spectral theory of measurable families of closed linear operators.