Advances in Mathematics AI1585 advances in mathematics 124, 7793 (1996) article no. 0078 Isomorphisms for Convergence Structures II. Borel Spaces and Infinite Measures Yeneng Sun Department of Mathematics, National University of Singapore, Republic of Singapore 119260 1. INTRODUCTION In [10] we introduced a new notion of isomorphisms for topological Viewmeasure metadata, spaces, citation whichand similar preserve papers almostat core.ac.uk sure continuity of mappings, almost brought to you by CORE sure convergence of random variables, and weak convergence of probabil- provided by Elsevier - Publisher Connector ity 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. 2. THE MAIN RESULTS As in [10], we denote the Borel algebra of a topological space X by BX . If + is a Borel measure on X, we call the pair (X, +) a topological measure space. In this paper we shall assume that any measure on a topological space which we work with is a Borel measure. A mapping f from (X, +) to a topological space Z is said to be +-continuous or continuous almost everywhere if the set of discontinuity points of f is contained in a Borel set B with +(B)=0. The measure + is said to be locally finite at a point x in X, if there is a neighborhood O of x such that +(O) is finite. If 77 0001-8708Â96 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. File: 607J 158501 . By:CV . Date:16:12:12 . Time:02:04 LOP8M. V8.0. Page 01:01 Codes: 3626 Signs: 2157 . Length: 50 pic 3 pts, 212 mm 78 YENENG SUN + is locally finite at every point in X, We simply say that it is locally finite. The Lebesque measure on the real line R and its restriction to any Borel subset of R is denoted by *. Now we shall present the definition of the isomorphism introduced and studied in [10]. Definition. For any given topological measure spaces (X, +) and (Y, &), a Borel measurable, measure preserving mapping F from X to Y is called an isomorphism if F is +-continuous and F &1 is &-continuous. We say that (X, +) is isomorphic to (Y, &) if there is an isomorphism from (X, +)to(Y,&). It follows from Theorem 1 of [10] that a Polish space with a finite non- atomic measure is isomorphic to a finite Lebesgue interval. On the other hand, any uncountable Borel space is Borel isomorphic to the unit interval [0, 1]. Thus it is natural to expect that a Borel space with a finite non- atomic measure is isomorphic to a finite Lebesgue interval. However, the following theorem and Example 2 indicate that it is in general not true. Theorem 1. Let & be a finite positive measure on a Borel space X with &(X)=b. Then (i) if & is nonatomic and there is a Polish space Y embedded in X such that &(X&Y)=0, then for any given countable set E in the intersection of the Polish space Y with the support of &, there is an isomophism F from (X, &) to ([0, b], *) such that F is continuous at every point in E and F&1 is continuous at every point in F(E); (ii) if (X, &) is isomorphic to ([0, b], *), then & is nonatomic and there is a Polish space Y embedded in X such that &(X&Y)=0. It is well known that any uncoutable Borel space contains a Cantor space (see [9], p. 11). So an immediate question is whether the condition in the above theorem that requires a Polish space to be contained in the Borel space with full measure is redundant, i.e. whether every Borel space with a finite nonatomic measure contains a Polish space with full measure. The following example answers the question in the negative. Example . 2 Let [rn]n=1 be a list of all rational numbers in I. Let X=[(rn , y): n1, y # I]. Then, as a subspace of R2, X is a Borel space. Every Borel subset of X can be expressed as n=1[rn]_An for some Borel sets An 's. Define a probabil- n ity measure + on X such that +(n=1[rn]_An)=n=1(1Â2 ) *(An). We shall show that X does not contain any Polish space with full measure. File: 607J 158502 . By:CV . Date:16:12:12 . Time:02:04 LOP8M. V8.0. Page 01:01 Codes: 3060 Signs: 2334 . Length: 45 pic 0 pts, 190 mm ISOMORPHISMS FOR CONVERGENCE STRUCTURES, II 79 Suppose X has a Polish subspace Y such that +(Y)=1. Let Y=n=1[rn]_Yn. Then *(Yn)=1 for each n. Thus *(n=1Yn)=1. Choose y0 # n=1Yn. Then the set A=[(rn , y0): n1], as a subspace of Y, is closed, which yields that A is also a Polish space. That is clearly impossible. Hence, by Theorem 1, we know that (X, +) is not isomorphic to (I, *). The next theorem characterizes those Borel spaces with finite measures (possibly with atoms) isomorphic to intervals with some finite measures. Theorem 3. Let & be a finite positive measure on a Borel space X. Let &(X)=b. Then the following two assertions are equivalent: (i) There exists a measure m on [0, b] with m([0, b])=b such that (X, &) is isomorphic to ([0, b], m). (ii) There is a Polish space Y embedded in X such that &(X&Y)=0 and every atom of & is a condensation point of X. Now we consider spaces with infinite measures. We characterize those Borel spaces with infinite measures isomorphic to the real line R with some measures or the Lebesgue measure. Theorem 4. Let + be a _-finite measure on a Borel space X with +(X)=. Then (i) The following two assertions are equivalent: (a) There exists a measure & on R such that (X, +) is isomorphic to (R, &). (b) There is a Polish space Y embedded in X such that +(X&Y)=0 and every atom x of + is a condensation point of X. (ii) The topological measure space (X, +) is isomophic to (R, *) if and only if + is nonatomic and there is a Polish space Y embedded in X such that +(X&Y)=0 and +| Y is locally finite. The following example exhibits that there are Polish spaces with _-finite infinite nonatomic measures which are nowhere locally finite. Example . 5 As in example 2, let [rn]n=1 be a list of all rational num- bers. Let X=[(rn , y): n1, y # R] define a _-finite measure + on R2 such that +(R2&X)=0 and +(n=1[rn]_An)=n=1*(An) for any Borel sets An 's. Then for any open File: 607J 158503 . By:CV . Date:16:12:12 . Time:02:04 LOP8M. V8.0. Page 01:01 Codes: 2741 Signs: 1883 . Length: 45 pic 0 pts, 190 mm 80 YENENG SUN set B in R2, +(B)=. Thus + is not locally finite at any point. Moreover, if F is a measure preserving mapping from (R2, +)to(R,*), then F is not continuous at any point. 3. THE PROOF Note that the completeness of the spaces studied in [10] is used quite freely there. Thus the method used in that paper cannot be adapted straightforwardly to prove the results in this paper. Actually, more intricate construcions have to be found for the new pupose. To show the existence of an isomorphism from a Borel space X with a finite measure & which satisfies the conditions in Theorem 3 (ii) to a finite interval with some measure, we can embed the Borel space in the Hilbert cube H as a Borel subset. Then by refining and introducing some new features into the constructions used in the proof of Theorem 1 of [10], we show that (X, &) is isomorphic to (H, &Ä ), where &Ä is the trivial extension of & to H. Then it follows from Theorem 2 of [10] that (X, &) is isomorphic to an interval with some measure. The existence of isomorphisms in other cases can be deduced from this result and Theorem 1 of [10] with relative ease. Now we fix some notation and introduce some basic definitions. For a metric space Y and a subset A of Y, let A%, A and A be the interior, closure and boundary of A respectively. Let d(A) be the diameter of A measured in the given metric of Y. Let Br(x) and Sr(x) be the open ball and the sphere centered at x with radius r respectively. The set A is said to be regular closed (regular open) if A%=A ((A )%=A). If A is open (closed), then it is clear that A (A%) is regular (regular open). Let E designate the set of all finite strings on the set [0, 1].