{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T23:57:42Z","timestamp":1648943862736},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":6128,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1997,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We give positive answers to two open questions from [15]. (1) For every set <jats:italic>C<\/jats:italic> countably determined over <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200016376_inline1\" \/>, if <jats:italic>C<\/jats:italic> is <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200016376_inline6\" \/> then it must be <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200016376_inline6\" \/> over <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200016376_inline1\" \/>, and (2) every Borel subset of the product of two internal sets <jats:italic>X<\/jats:italic> and <jats:italic>Y<\/jats:italic> all of whose vertical sections are <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200016376_inline6\" \/> 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 <jats:italic>X<\/jats:italic> is a measurable space and <jats:italic>Y<\/jats:italic> 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.<\/jats:p>","DOI":"10.2307\/2275548","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T23:01:20Z","timestamp":1146956480000},"page":"595-607","source":"Crossref","is-referenced-by-count":0,"title":["Louveau's theorem for the descriptive set theory of internal sets"],"prefix":"10.1017","volume":"62","author":[{"given":"Kenneth","family":"Schilling","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bo\u0161ko","family":"\u017divaljevi\u0107","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200016376_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/BF01387406"},{"key":"S0022481200016376_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(91)90010-J"},{"key":"S0022481200016376_ref010","volume-title":"Foundations of infinitesimal stochastic analysis","author":"Stroyan","year":"1986"},{"key":"S0022481200016376_ref009","volume-title":"Analytic sets","author":"Rogers","year":"1980"},{"key":"S0022481200016376_ref007","first-page":"363","article-title":"A separation theorem for  sets","volume":"260","author":"Louveau","year":"1980","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200016376_ref004","volume-title":"An introduction to nonstandard real analysis","author":"Hurd","year":"1985"},{"key":"S0022481200016376_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1993-1126195-9"},{"key":"S0022481200016376_ref002","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1979-066-0"},{"key":"S0022481200016376_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1982-0654856-1"},{"key":"S0022481200016376_ref008","unstructured":"[8] Render H. , private communication."},{"key":"S0022481200016376_ref013","first-page":"50","volume":"56","author":"\u017divaljevi\u0107","year":"1991","journal-title":"The structure of graphs all of whose Y-sections are internal sets"},{"key":"S0022481200016376_ref014","first-page":"172","volume":"57","author":"\u017divaljevi\u0107","year":"1992","journal-title":"Lusin-Sierpinski index for the internal sets"},{"key":"S0022481200016376_ref011","unstructured":"[11] \u017divaljevi\u0107 B. , functions are almost internal, to appear."},{"key":"S0022481200016376_ref005","first-page":"1167","volume":"54","author":"Keisler","year":"1989","journal-title":"Descriptive set theory over hyperfinite sets"},{"key":"S0022481200016376_ref006","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100000785"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200016376","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,11]],"date-time":"2019-05-11T20:55:40Z","timestamp":1557608140000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200016376\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,6]]},"references-count":15,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1997,6]]}},"alternative-id":["S0022481200016376"],"URL":"https:\/\/doi.org\/10.2307\/2275548","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1997,6]]}}}