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

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... lutely 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.