2022/04/03 by Mejak, Severin
#03E15 #03E75 #28A05 #28A33 #46L30 #52A05 #54H05 #FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2204.02767
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 MA + ¬ CH there are no Σ12 mofs, that under PD there are no projective mofs and that under 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.