The author proves the following generalization of a classical theorem of Kuratowski: Let \(X\) be a Polish space, \(f : X \to X\) be an injective Borel map and \(\mu\) be a nonatomic, \(\sigma\)-finite, positive measure on the Borel \(\sigma\)-field \({\mathcal B}\) of \(X\); let \({\mathcal B}_\mu\) be the \(\mu\)-completion of \({\mathcal B}\); in order that \(f(B) \in {\mathcal B}_\mu\) for every \(B \in {\mathcal B}_\mu\) it is necessary and sufficient that \(\mu \ll \mu_f = \mu f^{-1}\). Various related remarks are also given.