Given two internal sets X and Y we prove that every Borel function whose graph is a subset of the product X x Y is a member of the least set containing the class of all internal functions and closed with respect to the operations of monotone countable union and intersection. We also prove that any Souslin function can be extended to a Borel function and obtain, as a corollary, a new proof of the recent result of Henson and Ross (...) about the measure preserving property of injective Souslin functions. Further we give a new proof of the fact that injective Borel functions map Borel sets into Borel. A structural result for Souslin graphs all of whose Y-sections are of cardinality n is given. (shrink)

We prove that a Borel subset of the product of two internal setsX andY all of whoseY-sections are ∏ 1 0 (K)(∑ 1 0 (K)) sets is the intersection (union) of a countable sequence of Borel graphs with internalY-sections. As a consequence we prove some standard results about the domains of graphs in the product of two topological spaces all of whose horizontal section are compact (open) sets. A version of classical Vitali-Lusin theorem for those types of graphs is given (...) as well as a new proof (and an extension) of a classical result of Kunugui. (shrink)

We give positive answers to two open questions from [15]. (1) For every set C countably determined over A, if C is Π 0 α (Σ 0 α ) then it must be Π 0 α (Σ 0 α ) over A, and (2) every Borel subset of the product of two internal sets X and Y all of whose vertical sections are Π 0 α (Σ 0 α ) can be represented as an intersection (union) of Borel sets with (...) vertical sections of lower Borel rank. We in fact show that (2) is a consequence of the analogous result in the case when X is a measurable space and Y a complete separable metric space (Polish space) which was proved by A. Louveau and that (1) is equivalent to the property shared by the inverse standard part map in Polish spaces of preserving almost all levels of the Borel hierarchy. (shrink)

We prove that there exists a function f which reduces a given Π1 1 subset P of an internal set X of an ω1-saturated nonstandard universe to the set WF of well-founded trees possessing properties similar to those possessed by the standard part map. We use f to define the Lusin-Sierpinski index of points in X, and prove the basic properties of that index using the classical properties of the Lusin-Sierpinski index. An example of a Π1 1 but not Σ1 (...) 1 set is given. (shrink)

We prove that every countably determined set C is U-meager if and only if every internal subset A of C is U-meager, provided that the cofinality and coinitiality of the cut U are both uncountable. As a consequence we prove that for such cuts a countably determined set C which intersects every U-monad in at most countably many points is U-meager. That complements a similar result in [KL]. We also give some partial solutions to some open problems from [KL]. We (...) prove that the set K = {1,...,H}, where H is an infinite integer, cannot be expressed as a countable union of countably determined sets each of which is U-meager for some cut U with min{cf (U), ci (U)} ≥ ω1. Also, every Borel, Σ1 m or countably determined set C which is U-meager for every cut U is a countable union of Borel, Σ1 m or countably determined sets respectively, which are U-nowhere dense for every cut U. Further, the class of Borel U-meager sets for min{cf(U), ci(U)} ≥ ω1 coincides with the least family of sets containing internal U-meager sets and closed with respect to the operation of countable union and intersection. The same is true if the phrase "U-meager sets" is replaced by "U-meager for every cut U.". (shrink)