Orthogonality of measures and states

Abstract

We give a short proof of the theorem due to Preiss and Rataj stating that there are no analytic maximal orthogonal families (mofs) of Borel probability measures on a Polish space. When the underlying space is compact and perfect, we show that the set of witnesses to non-maximality is comeagre. Our argument is based on the original proof by Preiss and Rataj, but with significant simplifications. The proof generalises to show that under $\mathsf{MA} + \neg \mathsf{CH}$ there are no $\mathbf{\Sigma^1_2}$ mofs, that under $\mathsf{PD}$ there are no projective mofs and that under $\mathsf{AD}$ there are no mofs at all. We also generalise a result due to Kechris and Sofronidis, stating that for every analytic orthogonal family of Borel probability measures there is a product measure orthogonal to all measures in the family, to states on a certain class of C*-algebras.