Theorem of Kuratowski-Suslin for measurable mappings. (English) Zbl 0836.28003
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.
Reviewer: S.D.Chatterji (Lausanne)
MSC:
28A20 | Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence |
28C15 | Set functions and measures on topological spaces (regularity of measures, etc.) |
28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |
60B11 | Probability theory on linear topological spaces |