Measurable parametrizations and selections

Abstract

Abstract. Let W be a Borel subset of / X / (where / = [0, 1]) such that, for each x, Wx = {y: (x,y) e W} is uncountable. It is shown that there is a map, g, of I X I onto W such that (1) for each x, g(x, • ) is a Borel isomorphism of / onto Wx and (2) both g and g ~ ' are S(I X 7)-measurable maps. Here, if X is a topological space, S(X) is the smallest family containing the open subsets of X which is closed under operation (A) and complementation. Notice that S(X) is a subfamily of the universally or absolutely measurable subsets of X. This result answers a problem of A. H. Stone.This result improves a theorem of Wesley and as a corollary a selection theorem is obtained which extends the measurable selection theorem of von Neumann.We also show an analogous result holds if If is only assumed to be analytic.

Probability measures and theC-sets of Selivanovskij

“…This tf-field was studied by Selivanovskij [14] and is known to be contained in the α-field of universally measurable sets (see, e.g., Saks [13]). Furthermore, all the selection results mentioned above and even much stronger ones are true for S^(X) measurable functions [5]. We say that a mapping / from a Borel space X to a Borel space Y is S^{X) measurable if f~\B) e S^{X) for every Borel set BdY and show that this implies f~ι{B) e S^{X) for every B e This allows us to conclude that the composition of two measurable functions is again S^(X) measurable, which is the result of Lemma 2 of [5].…”

Some remarks about measurable parametrizations

“…Proof of Theorem 1.2. To make this note complete, we shall repeat a Schröder-Bernstein type argument due to Cenzer and Mauldin [CM,Proof of Theorem 4] to derive Theorem 1.2 from Corollary 2.2. Let B c 2" X 2" be a Borel set whose vertical sections B(x) are all uncountable, let k: 2" X 2" -» B be the map described in Corollary 2.2, let S0 = B\k(2" X2"), T0= (2" X 2U)\B, GO S" = k"(S0), Tn = k"(T0), D = fi *"(2" X 2"), « = i and let oc oo 77=7>U \js", G=Ur",…”

Some Applications of Selection Theorems to Parametrization Problems

“…Several authors, most notably Cenzer and Mauldin [1] and Mauldin [6], have shown that measurable parametrizations of Borel sets in the product of two Polish spaces can be constructed in two steps.…”