ISOMORPHISMS FOR CONVERGENCE STRUCTURES

For a nonatomic Borel probability measure mu on a Polish space X, an i somorphism from (X,mu) to the unit Lebesgue interval ([0, 1]), lambda) 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 som e structures involving convergence, which reveals to some extent the m ystery of convergence problems on relatively sophisticated spaces (for example, function spaces). Implications of such a result in ergodic t heory, probability theory, and probabilistic number theory are discuss ed. In particular, the study of generic orbits of an ergodic measure p reserving transformation on a general Polish space is equivalent to th e same problem on the unit Lebesgue interval. The fact that stochastic processes can be regarded as random elements of spaces of functions a llows us to claim that the convergence of a sequence of stochastic pro cesses is equivalent to the convergence of some sequence of random var iables taking values in the unit interval [0, 1]. Furthermore, the iso morphism studied in this paper preserves uniform distribution of seque nces. Thus many results in the abstract theory of uniform distribution can be obtained by transferring the corresponding results in the simp lest case of uniform distribution mod 1. (C) 1995 Academic Press, Inc.