The structure of graphs all of whose Y-sections are internal sets

Abstract

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]).

Graphs with ∏ 1 0 (K)Y-sections

“…It is an extension of a result, known before to many authors and used also in [13], which states that a set which is conntably determined over an algebra ,~ of internal sets and at the same time internal must actually be a member of the algebra ,~. It is an extension of a result, known before to many authors and used also in [13], which states that a set which is conntably determined over an algebra ,~ of internal sets and at the same time internal must actually be a member of the algebra ,~.…”

“…Proof The fact that the domain of a Borel graph with internal Y-sections is Borel is proved in [13] (Corollary 3.8 ii)). Now _F is either a union of an intersection of countably many Borel graphs with internal Y-sections.…”

“…In [13] (Proposition 2.1 ii)) we prove that every Borel graph _F with all of its Y-sections internal sets can be represented as a countable union of restrictions of internal graphs to Borel sets. The proof is completed.…”

“…In [13] this result has been extended to arbitrary Borel graphs all of whose Y-sections are internal sets. They proved that any function whose graph is a Borel subset of the product X • Y can be covered by countably many internal functions (provided that the nonstandard universe is cvl-saturated ).…”

“…In [13] this result has been extended to arbitrary Borel graphs all of whose Y-sections are internal sets. 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.…”