Measurable parametrizations and selections

Abstract:

Let W be a Borel subset of $I \times I$ (where $I = [0,1]$) such that, for each x, ${W_x} = \{ y: (x,y) \in W\}$ is uncountable. It is shown that there is a map, g, of $I \times I$ onto W such that (1) for each x, $g(x, \cdot )$ is a Borel isomorphism of I onto ${W_x}$ and (2) both g and ${g^{ - 1}}$ are $S(I \times I)$-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 W is only assumed to be analytic.