Measurable Choice Functions

Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/fun/choice functions.pdf] This note is essentially a rewrite of Appendix V of J. Dixmier’s von Neumann Algebras, North Holland, 1981. The point is to obtain the practical special case: [0.0.1] Corollary: Let X be a locally compact Hausdorff second-countable [1] topological space with a complete positive Borel measure ν giving compacta finite measure. Let Y be a complete second-countable metric space, Γ a Borel subset of X×Y . Let Xo be the projection of Γ to X. [2] Then there is a ν-measurable function f : Xo → Y whose graph lies inside Γ. ///