Orthogonality of measures and states
Volume 264 / 2024
Abstract
We give a short proof of the theorem due to Preiss and Rataj stating that there are no analytic maximal orthogonal families 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 $\mathbf{\Sigma }^1_2$ maximal orthogonal families, that under $\mathsf{PD}$ there are no projective maximal orthogonal families and that under $\mathsf{AD}$ there are no maximal orthogonal families 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.
