\((X, \mathcal{M}, \mu)\) is a finite measure space with associated outer measure \( \mu^{*}\): for any \(A \subset X, \mu^{*}(A) = \inf \{ \mu(B): B \in \mathcal{M}, B \supset A \}\), and \((Y, d_{Y})\) is a metric space. A function \( f: X \to Y\) is said to be \(\mathcal{M}\)-measurable if \( f^{-1} (U) \in \mathcal{M}\) for every open set \(U\) in \(Y\). For a sequence \( \{ f_{n}: X \to Y \}\) and an \( f: X \to Y\) (these functions are not necessarily measurable), \( f_{n}\) is defined to be convergent to \(f\) in outer measure if for any \( \varepsilon >0\), \( \mu^{*} (\{x: d_{Y}( f_{n}(x), f(x)) \geq \varepsilon\}) \to 0 \). Using this definition, the author extends the known results about preservation of convergence in measure, by taking composition, to the following main result of the paper: \( (X, \mathcal{M}, \mu)\) is a finite measure space, \((Y, d_{Y}), \; (Z, d_{Z}) \) are metric spaces, \( \{ f_{n}: X \to Y \}\) is a sequence of functions, \( \{ f: X \to Y \}\) an \(\mathcal{M}\)-measurable function, and \( \{ g: Y \to Z \}\) such that \(g\) is continuous on \(f(X)\). If \( f_{n}\) is convergent to \(f\) in outer measure, then \(g \circ f_{n}\) converges to \(g \circ f\) in outer measure.