Representing Projective Sets as Unions of Borel Sets

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 paper is to generalize Theorem 1 to higher levels of the projective hierarchy. The pointclass of 2^ sets is closed under well-ordered unions (Kechris, Solovay, and Steel [6, 2.4.1]), so clearly the higher level projective sets cannot be represented as a well-ordered union of Borel sets. But they will be represented by a special kind of union which characterizes the pointclass 2d2n+2 ■ Let X denote a projective ordinal. Let px denote the supercompact measure on PWl(X) which is defined in Becker [1]. (Woodin [9] has shown that px is, in fact, the only supercompact measure on PW] (X).) We assume familiarity with the basic facts about px, all of which can be found in Becker [1]. We also assume familiarity with the theory of projective sets, under AD, as presented in Moschovakis [8]. Let 3§ denote the class of Borel subsets of cow . Definition. Let A c cow, and let F : Pw¡ (X) —> 3§ . F is called a X-representation of A if for any set 5e c PWi (X) such that Px(^) = 1, A = u F(S). We say that A is X-representable if there exists a A-representation of A . Proposition 2. Let A c cou>. (a) Let F : PW](X) —> 3S . F is a X-representation of A iff both of the following properties hold. (i) For all S £ P(ûl(X),F(S) c A. (ii) For all x £ co03, if x £ A, then for px-a.e. S,x £ F(S). (b) A is unrepresentable iff A is the union of cox Borel sets. (c) A is oi-representable iff A is Borel. Proof, (b) poj^-a.e. set in PW](cox) is an initial segment of cox. D Received by the editors May 27, 1993. 1991 Mathematics Subject Classification. Primary 03E15, 03E60. Partially supported by National Science Foundation Grant DMS-9206922. © 1995 American Mathematical Society 0002-9939/95 $1.00+ $.25 per page