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.In this note we examine to what extent convergence in measure for a sequence of measurable mappings is preserved under composition by another measurable mapping. More precisely, if (f n ) is a sequence of measurable mappings such that (f n ) converges to f in measure (we will review the pertinent definitions shortly), and if g is a measurable mapping, then under what conditions can we infer that (g • f n ) converges to g • f in measure? This is a natural question which has been around for some time now (see, for example, [1]). Positive answers to this question have useful applications in probability, statistics, and stochastic processes because convergence in measure (also called convergence in probability) is a basic mode of convergence for sequences of random variables (for specific applications, see, for example, [3, p. 64] and [8, pp. 104, 259]).