Measurable cross sections (Q801429)

The main result of this paper is that, if the canonical map from the hyperstonean of the compact measure space (X,m) to X has an m-Borel- measurable cross section, then every continuous map from Y onto X (Y is any compact space) has an m-Borel-measurable cross section (by using Gleason's theorem on projective topological spaces). From this result follows as a corollary the well known result of Kupka, that every continuous map from a compact Y onto X has an m-Borel-measurable cross sections, if (X,m) has the strong lifting property. Remark: The Proposition 2 of this paper is connected also with results of the reviewer [for example in Math. Ann. 197, 279-285 (1972; Zbl 0223.28014)].