Isomorphisms for Convergence Structures. II. Borel Spaces and Infinite Measures

In [10] we introduced a new notion of isomorphisms for topological measure spaces, which preserve almost sure continuity of mappings, almost sure convergence of random variables, and weak convergence of probability measures. The main thrust of that paper is the construction of an isomorphism from a Polish space with a nonatomic Borel probability measure to the unit Lebesgue interval (I, *). Applications in ergodic theory, probability theory, and probabilistic number theory are also discussed there. The purpose of this paper is to continue the study along that line. In particular, as suggested by the classic Borel Isomorphism Theorem (see [9]), we shall consider general Borel spaces in this paper instead of Polish spaces in [10]. Here a Borel space is a topological space homeomorphic to a Borel subset of a complete separable metric space with the subspace topology. In ergodic theory and the theory of Markov chains, infinite invariant measures have also been studied (see [1, 5, 6, 7] and many others). To establish a connection between convergence problems on the real line, we shall also provide isomorphism results for Borel spaces with infinite measures.