The purpose of this paper is to give structural results on graphs lying in the product of two hyperfinite sets X and Y, whose Y-sections are either all internal sets or all of “small” cardinality with respect to the saturation assumption imposed on our nonstandard universe. These results generalize those of [KKML] and [HeRo]. In [KKML] Keisler, Kunen, Miller and Leth proved, among other results, that any countably determined function in the product of two internal sets X and Y can be covered by countably many internal functions provided that the nonstandard universe is at least ℵ-saturated. This shows that any countably determined function can be represented as a union of countably many restrictions of internal functions to countably determined sets. On the other hand, Henson and Ross use in [HeRo] Choquet's capacitability theorem to prove that any Souslin function in the product of two internal sets X and Y is a.e. equal to an internal function. (Here “a.e.” refers to an arbitrary but fixed bounded Loeb measure.) Therefore, in our terminology, every Souslin function possesses an internal a.e. lifting.After the introductory §0, where all the necessary terminology is introduced, we continue by presenting the structural result for graphs all of whose Y-sections are of cardinality ≤κ (provided that the nonstandard universe is ≤κ+-saturated) in §1. We show that, under the above saturation assumption, a κ-determined graph with all of the Y-sections of cardinality ≤κ is covered by κ-many internal functions. Therefore, any such graph is a union of κ-many κ-determined functions. In particular if the graph in question is Borel, Souslin, κ-Borel or κ-Souslin (or a member of one of the Borel, κ-Borel or projective hierarchies) then the corresponding constituting functions are of the same “complexity”. Thus, any Borel graph all of whose Y-sections are at most countable is a union of countably many Borel functions and, consequently, has Borel domain. In the setting of Polish topological spaces this was proved by Novikov (see [De]).

A remarkable result of Henson and Ross [HR] states that if a function whose graph is Souslin in the product of two hyperfinite sets in an ℵ1 saturated nonstandard universe possesses a certain nice property (capacity) then there exists an internal subfunction of the given one possessing the same property. In particular, they prove that every 1-1 Souslin function preserves any internal counting measure, and show that every two internal sets A and B with ∣A∣/∣B∣ ≈ 1 are Borel bijective. As a supplement to the last-mentioned result of Henson and Ross, Keisler, Kunen, Miller and Leth showed [KKML] that two internal sets A and B are bijective by a countably determined bijection if and only if ∣A∣/∣B∣ is finite and not infinitesimal.In this paper we first show that injective Borel functions map Borel sets into Borel sets, a fact well known in classical descriptive set theory. Then, we extend the result of Henson and Ross concerning the Borel bijectivity of internal sets whose quotient of cardinalities is infinitely closed to 1. We prove that two Borel sets, to which we may assign a counting measure not equal to 0 or ∞, are Borel bijective if and only if they have the same counting measure ≠0, ∞. This, together with the similar characterization for Souslin and measurable countably determined sets, extends the above-mentioned results from [HR] and [KKML].

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AbtractWe 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 , 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, or countably determined set C which is U-meager for every cut U is a countable union of Borel, 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.”