Mapping properties that preserve convergence in measure on finite measure spaces

Abstract

Given a finite measure space ( X , M , μ ) and given metric spaces Y and Z, we prove that if { f n : X → Y | n ∈ 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 ○ f n 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.