A measure-Theoretic Wagde Hierarchy

Benedikt Löwe

We shall look at a relation ≤_{BlW} on sets of reals that is closely related to Wadge reducibility (cf. [4] and [1]). For two sets of reals A and B we say that A ≤_{BlW} B holds if and only if the Wadge game G_{W}(A, B) has probabilistic value 0 (cf. [2]). Of course, this is a weaker reducibility relation that Wadge reducibility in the sense that A ≤_{W} B implies A ≤_{BlW} B. By a theorem of Neeman (cf. [3]), the two relations have to coincide under the assumption of Blackwell Determinacy throughout the L(ℝ) hierarchy. It is an open question whether they are equal everywhere – this question is equivalent to “Does Blackwell Determinacy imply Wadge Determinacy?”.

In this talk we shall show (assuming Blackwell Determinacy) that the relation ≤_{BlW} exhibits the very same structural properties as ≤_{W} under AD. They can only fail to be the same if ≤_{W} is not wellfounded, i.e., if Wadge Determinacy is false.

[1] Alessandro Andretta, The Secret Lore of the Cabal, monograph in preparation

[2] Donald A. Martin, The Determinacy of Blackwell Games, Journal of Symbolic Logic 63 (1998), p. 1565–1581

[3] Donald A. Martin, Itay Neeman, The Strength of Blackwell Determinacy, in preparation

[4] Robert van Wesep, Wadge Degrees and Descriptive Set Theory, in: Alexander S. Kechris, Yiannis N. Moschovakis (eds.), Cabal Seminar 76–77, Proceedings, Caltech–UCLA Logic Seminar 1976–77, Berlin 1978 [Lecture Notes in Mathematics 689], p. 151–170

Address: Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Beringstraße 6, D—53115 Bonn, Germany

E-mail: loewe@math.uni-bonn.de