1 Probability measures on metric spaces 1 1.1 Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Borel probability measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Narrow convergence of measures . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 The bounded Lipschitz metric . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5...

We consider a method of representing projective sets by a particular type of union of Borel sets, assuming AD. We prove a generalization of the theorem that a set is S^ iff it is the union of a>x Borel sets. The Axiom of Determinacy (AD) is always assumed. Theorem 1 (Sierpinski, Solovay, Moschovakis [8, 7D.10]). Let A c com. A is Ej iff A is the union of cox Borel sets. The purpose of this pape...

Given a countable Borel equivalence relation E on a Polish space, let IE denote the σ-ideal generated by the Borel partial transversals of E. We show that there is a Borel homomorphism from IE to IF if and only if there is a smooth-to-one Borel homomorphism from a finite index Borel subequivalence relation of E to F . As a corollary, we see that IE is homogeneous in the sense of Zapletal (2007,...

We present several selection theorems for Borel relations, involving only Borel sets and functions, all of which can be obtained as consequences of closely related theorems proved in [DSR 96,99,01,01X] involving coanalytic sets. The relevant proofs given there use substantial set theoretic methods, which were also shown to be necessary. We show that none of our Borel consequences can be proved ...

See [Oli04]. In [AdKech00], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was...

2. Proposition. For all n ∈ ω there exists a subset B of ω such that E n 6= E n+1. This proposition follows from a theorem of M. Sipser about Boolean circuits (see “Borel Sets and Circuit Complexity”, JACM 1983, pp. 61–69), whose presentation we closely follow. On the way, we study a theorem of Ajtai about the structure of classes of sets of the form E n that uses an analogue of the Borel hiera...

The literature provides dichotomies involving homomorphisms (like the G0 dichotomy) or reductions (like the characterization of sets potentially in a Wadge class of Borel sets, which holds on a subset of a product). However, part of the motivation behind the latter result was to get reductions on the whole product, like in the classical notion of Borel reducibility considered in the study of an...

This paper is a continuation of the study of topological properties of omega context free languages (ω-CFL). We proved in [Topological Properties of Omega Context Free Languages, Theoretical Computer Science, Volume 262 (1-2), 2001, p. 669697] that the class of ω-CFL exhausts the finite ranks of the Borel hierarchy, and in [Borel Hierarchy and Omega Context Free Languages, Theoretical Computer ...