Asger Törnquist: Definable maximal orthogonal families in forcing extensions

Place: Fields Institute (Room 210)
Date: 12-June-2015 (13:30-15:00)
Speaker: Asger Törnquist
Title: Definable maximal orthogonal families in forcing extensions.Abstract: Two Borel probability measures nu and mu on Cantor space are orthogonal if there is a Borel set which has measure 1 for nu, but measure 0 for mu. An orthogonal family of measures is a family of pairwise orthogonal measures; it is maximal if it is maximal under inclusion.

Maximal orthogonal families of measures can’t be analytic; this is a
theorem of Preiss and Rataj (1985). A few years ago, Vera Fischer and I showed that in L there is a Pi-1-1 (lightface) maximal orthogonal family (a “mof”) of measures in L, but that adding a Cohen real to L destroys all Pi-1-1 mofs. Subsequently, it was shown that the same holds if we add a random real (Friedman-Fischer-T.).

This motivated the question: Can a Pi-1-1 mof coexist with a
non-constructible real? In this talk we answer this by showing there is a Pi-1-1 mof in the Sacks and Miller extensions of L. By contrast, we will see that in the Mathias extension of L there are no Pi-1-1 mofs, and in the process of doing so we will obtain a new proof of the Preiss-Rataj theorem.  This is joint work with David Schrittesser.

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