Isomorphisms for convergence structures
The paper is a continuation of ideas of Kuratowski and von Neumann [cf. \textit{J. von Neumann}, Ann. Math., II. Ser. 33, 574-586 (1932; Zbl 0005.05603)]. Let \((X,\mu)\), \((Y,\nu)\) be topological measure spaces and let \(F:X\to Y\) be a Borel measurable, measure-preserving, bijective mapping. The function \(F\) is said to be an isomorphism provided that \(F\) is \(\mu\)-continuous and \(F^{-1}\) is \(\nu\)-continuous. The space \((X,\mu)\) is said to be isomorphic to \((Y,\nu)\) if there exists an isomorphism \(F:X\to Y\). The main result: Let \(\mu\) be a non-atomic Borel probability measure on the Polish space \(X\) (\(X\) is homeomorphic to a complete separable metric space). Let \(D\) be a countable set in the support of \(\mu\). Then there exists an isomorphism \(F\) from \((X,\mu)\) to \((I,\lambda)\) (\(I\) is the unit interval, \(\lambda\) is the Lebesgue measure) such that \(F\) is continuous at every point in \(D\) and \(F^{-1}\) is continuous at every point in \(F(D)\). Corollary: Let \((X,\mu)\) have the previous meaning. Then there exists a subset \(Y\) of \(I\) and a Borel probability measure \(\nu\) on \(Y\) such that \((X,\mu)\) is isomorphic to \((Y,\nu)\). Many examples and applications are given mainly in general theory of uniform distribution of sequences.
scientific article; zbMATH DE number 863625