Measurable parametrizations of sets in product spaces

Abstract

Abstract. Various parametrization theorems are proved. In particular the following is shown: Let B be a Borel subset of / X / (where / = [0,1]) with uncountable vertical sections. Let 2 UN be the discrete (topological) union of 2, the space of irrationals, and N, the set of natural numbers with discrete topology. Then there is a map /: / X (2 U N) -* F measurable with respect to the product of the analytic a-field on / (that is, the smallest a-field on / containing the analytic sets) and the Borel o-field on 2 U N such that/(f, ■): 2 U N -* / is a one-one continuous map of 2 U TV onto [x: (t, x) G B) for each t G T. This answers a question of Cenzer and Mauldin. 0. Introduction. Let (T, <31t) be a measurable space and X a Polish space. Suppose B E 9H ® %x, where 9>x is the Borel a-field of X, such that each vertical section of B is uncountable. In this article we consider the following kind of "parametrization" problem: Find a map /: T X Y -» X, where 7 = 2, the space of irrationals, or Y = 2 U N, the discrete (topological) union of 2 and N, the set of natural numbers with discrete topology, such that/(i, •) is a one-one continuous function on Y onto B' and/has suitable measurability properties. We then call/a one-one Carathéodory map and Y the parametrizing space. Several authors have considered this kind of problem and an extensive bibliography is to be found in Wagner [14].We begin, in §2, with parametrizations of certain kinds of measurable GÄ-valued multifunctions. The basic result is a "uniform" version of Mazurkiewicz's theorem that a dense Gs subset of a zero-dimensional Polish space whose complement is also dense is a homeomorph of 2, that is, we obtain here a Borel .measurable /: 7 X 2 -» X such that /(/, •) is a homeomorphism onto B' for each t E T. We use this technique to obtain one-one Carathéodory representations for measurable dense-in-itself Gs-valued multifunctions. These results on GÄ-valued multifunctions are used in the sequel. We might add that R. D. Mauldin and H. Sarbadhikari have independently obtained similar results [8].In §3 we solve a question posed by Cenzer and Mauldin [3] on the parametrization of Borel subsets B of I X I with uncountable sections where / is the unit interval. Wesley [15] had earlier taken the first step in this problem when he showed, using forcing techniques, that there is an Immeasurable map /: I X / -» /, where £2 is the Lebesgue a-field on / X /, such that/(/, •) is a Borel isomorphism on / onto B' for