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.