Isomorphisms for Convergence Structures

Abstract

For a nonatomic Borel probability measure μ on a Polish space X, an isomorphism from ( X, μ) to the unit Lebesgue interval ([0, 1]), λ) is constructed such that weak convergence of measures and almost sure convergence of random variables are preserved. Thus the unit interval together with the Lebesgue measure has a sort of universality for some structures involving convergence, which reveals to some extent the mystery of convergence problems on relatively sophisticated spaces (for example, function spaces). Implications of such a result in ergodic theory, probability theory, and probabilistic number theory are discussed. In particular, the study of generic orbits of an ergodic measure preserving transformation on a general Polish space is equivalent to the same problem on the unit Lebesgue interval. The fact that stochastic processes can be regarded as random elements of spaces of functions allows us to claim that the convergence of a sequence of stochastic processes is equivalent to the convergence of some sequence of random variables taking values in the unit interval [0, 1]. Furthermore, the isomorphism studied in this paper preserves uniform distribution of sequences. Thus many results in the abstract theory of uniform distribution can be obtained by transferring the corresponding results in the simplest case of uniform distribution mod 1.