Π¹₁ functions are almost internal

Abstract

Abstract.In Analytic mappings on hyperfinite sets [Proc. Amer. Math. Soc. (1993), 587-596] Henson andRoss asked for what hyperfinite sets S and T does there exists a bijection f from 5 onto T whose graph is a projective subset of S x r? In particular, when is there a nj bijection from S onto 77 In this paper we prove that given an internal, bounded measure p, any nj function is L(p) a.e. equal to an internal function, where L(p) is the Loeb measure associated with p. It follows that if two nj subsets S and T of a hyperfinite set X are nj bijective, then S and T have the same measure for every uniformly distributed counting measure p.. When 5 and T are internal it turns out that any nj bijection between them must already be Borel. We also prove that if a nj graph in the product of two hyperfinite sets X and Y is universal for all internal subsets of Y , then \X\ > 2lyl , which is a partial answer to Henson and Ross's Problem 1.5. At the end we prove some standard results about the projections and a structure of co-proper if-analytic subsets of the product of two completely regular HausdorfF topological spaces with open vertical sections. We were able to prove the above results by revealing the structure of n! subsets of the products X x Y of two internal sets X and y, all of whose F-sections are Z?(íc) sets.