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.
A preference order, or linear preorder, on a set X is a binary relation which is transitive, reflexive and total. This preorder partitions the set X into equivalence classes of the form . The natural relation induced by on the set of equivalence classes is a linear order. A well-founded preference order, or prewellordering, will similarly induce a well-ordering. A representation or Paretian utility function of a preference order is an order-preserving map f from X into the R of real numbers (provided with the standard ordering). Mathematicians and economists have studied the problem of obtaining continuous or measurable representations of suitably defined preference orders [4, 7]. Parametrized versions of this problem have also been studied [1, 7, 8]. Given a continuum of preference orders which vary in some reasonable sense with a parameter t, one would like to obtain a continuum of representations which similarly vary with t.
1. Introduction. The concept of inductive definability has become of great interest to recursion theorists in recent years. Recursion over natural numbers, ordinals, and higher type objects may itself be defined by an inductive operator-see for example [7] and [9]. Many results have been obtained characterizing the closures of inductive operators over the natural numbers, and relating lengths of inductive definitions to various interesting ordinals ; see [3] for a brief summary.The purpose of this note is to present results on the closure ordinals and sets of inductive operators over the continuum. Details will appear later in
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