Mapping properties that preserve convergence in measure on finite measure spaces

摘要

Given a finite measure space (X, M, mu) and given metric spaces Y and Z, we prove that if {f(n) : X -> Y vertical bar n is an element of N} is a sequence of arbitrary mappings that converges in outer measure to an M-measurable mapping f : X -> Y and if g: Y -> Z is a mapping that is continuous at each point of the image of f, then the sequence g o f(n) converges in outer measure to g o f. We must use convergence in outer measure, as opposed to (pure) convergence in measure, because of certain set-theoretic difficulties that arise when one deals with nonseparably valued measurable mappings. We review the nature of these difficulties in order to give appropriate motivation for the stated result. (c) 2006 Elsevier Inc. All rights reserved.