Analytic mappings on hyperfinite sets

Abstract

Let S S and T T be hyperfinite sets in an ℵ 1 {\aleph _1} -saturated nonstandard universe. The following are equivalent: (i) | S | | T | ≈ 1 \frac {{|S|}} {{|T|}} \approx 1 . (ii) There is a bijection from | S | | T | ≈ 1. \frac {{|S|}} {{|T|}} \approx 1. onto T T whose graph is Borel (over the internal subsets of S × T S \times T ). This follows from somewhat more general results about analytic partial functions on hyperfinite sets, the proofs of which use Choquet’s theorem on the capacitibility of analytic sets. This paper includes: proofs of the above results; an elementary direct construction for extensions of internal set functions to capacities; and a surprising corollary asserting the nonexistence of ergodic Borel transformations of a Loeb probability space.