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 is a sequence of arbitrary mappings that converges in outer measure to an M-measurable mapping and if 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.