$\Pi ^ 1_ 1$ functions are almost internal

Abstract:

In Analytic mappings on hyperfinite sets [Proc. Amer. Math. Soc. 2 (1993), 587-596] Henson and Ross asked for what hyperfinite sets $S$ and $T$ does there exists a bijection $f$ from $S$ onto $T$ whose graph is a projective subset of $S \times T$? In particular, when is there a $\Pi _1^1$ bijection from $S$ onto $T$? In this paper we prove that given an internal, bounded measure $\mu$, any $\Pi _1^1$ function is $L(\mu )$ a.e. equal to an internal function, where $L(\mu )$ is the Loeb measure associated with $\mu$. It follows that if two $\Pi _1^1$ subsets $S$ and $T$ of a hyperfinite set $X$ are $\Pi _1^1$ bijective, then $S$ and $T$ have the same measure for every uniformly distributed counting measure $\mu$. When $S$ and $T$ are internal it turns out that any $\Pi _1^1$ bijection between them must already be Borel. We also prove that if a $\Pi _1^1$ graph in the product of two hyperfinite sets $X$ and $Y$ is universal for all internal subsets of $Y$, then $|X| \geqslant {2^{|Y|}}$, 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 $K$-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 $\Pi _1^1$ subsets of the products $X \times Y$ of two internal sets $X$ and $Y$, all of whose $Y$-sections are $\Sigma _1^0(\kappa )$ sets.