Mapping properties that preserve convergence in measure on finite measure spaces

Given a finite measure space (X,M,μ) and given metric spaces Y and Z, we prove that if {fn :X→ Y | n ∈ N} is a sequence of arbitrary mappings that converges in outer measure to anM-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 ◦ fn converges in outer measure to g ◦ 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. © 2006 Elsevier Inc. All rights reserved.