{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T08:44:44Z","timestamp":1775637884873,"version":"3.50.1"},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8412,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1991,3]]},"abstract":"<jats:p>The purpose of this paper is to give structural results on graphs lying in the product of two hyperfinite sets <jats:italic>X<\/jats:italic> and <jats:italic>Y<\/jats:italic>, whose <jats:italic>Y<\/jats:italic>-sections are either all internal sets or all of \u201csmall\u201d 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 <jats:italic>X<\/jats:italic> and <jats:italic>Y<\/jats:italic> can be covered by countably many internal functions provided that the nonstandard universe is at least \u2135-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 <jats:italic>X<\/jats:italic> and <jats:italic>Y<\/jats:italic> is a.e. equal to an internal function. (Here \u201ca.e.\u201d refers to an arbitrary but fixed bounded Loeb measure.) Therefore, in our terminology, every Souslin function possesses an internal a.e. lifting.<\/jats:p><jats:p>After the introductory \u00a70, where all the necessary terminology is introduced, we continue by presenting the structural result for graphs all of whose <jats:italic>Y<\/jats:italic>-sections are of cardinality \u2264<jats:italic>\u03ba<\/jats:italic> (provided that the nonstandard universe is \u2264<jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>-saturated) in \u00a71. We show that, under the above saturation assumption, a <jats:italic>\u03ba<\/jats:italic>-determined graph with all of the <jats:italic>Y<\/jats:italic>-sections of cardinality \u2264<jats:italic>\u03ba<\/jats:italic> is covered by <jats:italic>\u03ba<\/jats:italic>-many internal functions. Therefore, any such graph is a union of <jats:italic>\u03ba<\/jats:italic>-many <jats:italic>\u03ba<\/jats:italic>-determined functions. In particular if the graph in question is Borel, Souslin, <jats:italic>\u03ba<\/jats:italic>-Borel or <jats:italic>\u03ba<\/jats:italic>-Souslin (or a member of one of the Borel, <jats:italic>\u03ba<\/jats:italic>-Borel or projective hierarchies) then the corresponding constituting functions are of the same \u201ccomplexity\u201d. Thus, any Borel graph all of whose <jats:italic>Y<\/jats:italic>-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]).<\/jats:p>","DOI":"10.2307\/2274903","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:38:25Z","timestamp":1146955105000},"page":"50-66","source":"Crossref","is-referenced-by-count":6,"title":["The structure of graphs all of whose <i>Y<\/i>-sections are internal sets"],"prefix":"10.1017","volume":"56","author":[{"given":"Bo\u0161ko","family":"\u017divaljevi\u0107","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200024890_ref007","first-page":"1","volume-title":"Analytic sets","author":"Jayne","year":"1980"},{"key":"S0022481200024890_ref013","first-page":"604","volume":"55","author":"\u017divaljevi\u0107","year":"1990","journal-title":"Some results about Borel sets in descriptive set theory of hyperfinite sets"},{"key":"S0022481200024890_ref011","volume-title":"Foundations of infinitesimal stochastic analysis","author":"Stroyan","year":"1986"},{"key":"S0022481200024890_ref005","unstructured":"Henson C. W. and Ross D. , Analytic mappings on hyperfinite sets (to appear)."},{"key":"S0022481200024890_ref006","volume-title":"An introduction to nonstandard real analysis","author":"Hurd","year":"1985"},{"key":"S0022481200024890_ref014","unstructured":"\u017divaljevi\u0107 B. , Every Borel function is monotone Borel (to appear)."},{"key":"S0022481200024890_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1979-0521888-9"},{"key":"S0022481200024890_ref001","doi-asserted-by":"crossref","first-page":"149","DOI":"10.4064\/fm-107-2-149-159","article-title":"Borel sets with F\u03c3\u03b4 sections","volume":"107","author":"Bourgain","year":"1980","journal-title":"Fundamenta Mathematicae"},{"key":"S0022481200024890_ref008","first-page":"1167","volume":"54","author":"Keisler","year":"1989","journal-title":"Descriptive set theory over hyperfinite sets"},{"key":"S0022481200024890_ref002","first-page":"183","volume-title":"Analytic sets","author":"Dellacherie","year":"1980"},{"key":"S0022481200024890_ref009","volume-title":"Descriptive set theory","author":"Moschovakis","year":"1980"},{"key":"S0022481200024890_ref010","doi-asserted-by":"publisher","DOI":"10.24033\/bsmf.1835"},{"key":"S0022481200024890_ref012","volume-title":"Introduction to the theory of infinitesimals","author":"Stroyan","year":"1976"},{"key":"S0022481200024890_ref003","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1979-066-0"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200024890","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T22:36:17Z","timestamp":1558132577000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200024890\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,3]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1991,3]]}},"alternative-id":["S0022481200024890"],"URL":"https:\/\/doi.org\/10.2307\/2274903","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,3]]}}}