Analytic mappings on hyperfinite sets

Abstract:

Let $S$ and $T$ be hyperfinite sets in an ${\aleph _1}$-saturated nonstandard universe. The following are equivalent: (i) $\frac {{|S|}} {{|T|}} \approx 1$. (ii) There is a bijection from $\frac {{|S|}} {{|T|}} \approx 1.$ onto $T$ whose graph is Borel (over the internal subsets of $S \times T$). 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.