Isomorphisms for convergence structures. II: Borel spaces and infinite measures (Q1354660)

This paper is a continuation of an earlier author's paper [[1] Adv. Math. 116, No. 2, 322-355 (1995; Zbl 0867.28003)], in which a new notion of isomorphism is introduced. In [1] a construction of such isomorphism from a Polish space with nonatomic Borel measure to the unit interval with Lebesgue measure is given. In the present paper the considerations from [1] are extended to general Borel spaces (instead of Polish spaces). A topological space is called a Borel space if it is homeomorphic to a Borel subset of a complete separable metric space. Let \((X,\mu)\), \((Y,\nu)\) be two topological measure spaces. A Borel measurable, measure-preserving mapping \(F:X\to Y\) is called an isomorphism if \(F\) is \(\mu\)-continuous and \(F^{-1}\) is a \(\nu\)-continuous mapping. If such an isomorphism \(F: X\to Y\) exists, then \((X,\mu)\) is said to be isomorphic to \((Y,\nu)\). A typical result: Let \(\nu\) be a finite positive measure on a Borel space \(X\). Put \(\nu(X)= b\). Then the following two assertions are equivalent: (a) There exists a measure \(\mu\) on \([0, b]\) with \(\mu([0, b])= b\) such that \((X,\nu)\) is isomorphic to \(([0,b],\mu)\). (b) There is a Polish space \(Y\) embedded in \(X\) such that \(\nu(X\setminus Y)= 0\) and every atom of \(\nu\) is a condensation point of \(X\).