Maximal sets of orthogonal measures are not analytic

Abstract

By proving the theorem given in the title we answer a question posed by D. Mauldin at the conference on Measure Theory held in Oberwolfach in 1981. In this note we study maximal sets of pairwise orthogonal probability measures on (0,1). The existence of such sets is guaranteed by the Kuratowski-Zorn principle. Our theorem says that such sets cannot be Borel in the weak topology, or even analytic. This answers a problem posed by D. Mauldin at the Conference on Meas.ure Theory held in Oberwolfach in 1981 (see (2)). To study maximal sets of pairwise orthogonal measures is useful in the light of general properties of orthogo- nal kernels; see (2) for a survey of these results and further references. First let us introduce some notation. Assume that K is a compact metric space. Ji(K) denotes the locally convex space of all finite Borel signed measures on K with weak topology. Jf+(K) denotes its subspace of all positive measures and SP(K) its subspace of all probability measures. It is well known that @(K) is a compact metrizable space. If x G K we denote by ev the Dirac probability measure concentrated at x. Let 2(K) denote the set of all finite convex combinations of Dirac probabilities on K.