Definable maximal discrete sets in forcing extensions

Let R be a Σ ¹ ₁ binary relation, and recall that a set A is R-discrete if no two elements of A are related by R. We show that in the Sacks and Miller forcing extensions of L there is a ∆ ¹ ₂ maximal R-discrete set. We use this to answer in the negative the main question posed in [7] by showing that in the Sacks and Miller extensions there is a Π ¹ ₁ maximal orthogonal family (“mof”) of Borel probability measures on Cantor space. By contrast, we show that if there is a Mathias real over L then there are no Σ ¹ ₂ mofs.