Regularity properties of functional equations and inequalities
Identifieur interne : 001999 ( Istex/Corpus ); précédent : 001998; suivant : 001A00Regularity properties of functional equations and inequalities
Auteurs : Karl-Goswin Grosse-ErdmannSource :
- aequationes mathematicae [ 0001-9054 ] ; 1989-06-01.
Abstract
Summary: By a well-known theorem of Lebesgue and Fréchet every measurable additive real function is continuous. This result was improved by Ostrowski who showed that a (Jensen-) convex real function must be continuous if it is bounded above on a set of positive Lebesgue measure. Recently, R. Trautner provided a short and elegant proof of the Lebesgue—Fréchet theorem based on a representation theorem for sequences on the real line. We consider here a locally compact topological groupX with some Haar measure. Then the following generalizes Trautner's theorem: Theorem.Let M be a measurable subset of X of positive finite Haar measure. Then there is a neighbourhood W of the identity e such that for each sequence (z n )in W there is a subsequence (z nk )and points y and x k in M with z nk =x k ·y −1 for k ∈ℕ. Using this theorem we obtain the following extensions of the theorems of Lebesgue and Fréchet and of Ostrowski. Theorem.Let R and T be topological spaces. Suppose that R has a countable base and that X is metrizable. If g: X → R and H: R × X → T are mappings where g is measurable on a set M of positive finite Haar measure and H is continuous in its first variable, then any solution f: X → T of f(x · y) = H(g)(x), y) for x, y∈X is continuous. Theorem.Let G: X × X → ℝ be a mapping. If there is a subset M of X of positive finite Haar measure such that for each y∈X the mapping x ↦ G(x, y) is bounded above on M, then any solution f: x → ℞ of f(x · y) ⩽ G(x, y) for x, y∈X is locally bounded above. We also prove category analogues of the above results and obtain similar results for general binary mappings in place of the group operation in the argument off.
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DOI: 10.1007/BF01836446
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Regularity properties of functional equations and inequalities</title><author><name sortKey="Grosse Erdmann, Karl Goswin" sort="Grosse Erdmann, Karl Goswin" uniqKey="Grosse Erdmann K" first="Karl-Goswin" last="Grosse-Erdmann">Karl-Goswin Grosse-Erdmann</name><affiliation><mods:affiliation>Fachbereich IV—Mathematik, Universität Trier, Postfach 3825, D-5500, Trier, West Germany</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:AF5109B4D456E24567006A47CE18213D345CB6B7</idno><date when="1989" year="1989">1989</date><idno type="doi">10.1007/BF01836446</idno><idno type="url">https://api.istex.fr/document/AF5109B4D456E24567006A47CE18213D345CB6B7/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">001999</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">001999</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Regularity properties of functional equations and inequalities</title><author><name sortKey="Grosse Erdmann, Karl Goswin" sort="Grosse Erdmann, Karl Goswin" uniqKey="Grosse Erdmann K" first="Karl-Goswin" last="Grosse-Erdmann">Karl-Goswin Grosse-Erdmann</name><affiliation><mods:affiliation>Fachbereich IV—Mathematik, Universität Trier, Postfach 3825, D-5500, Trier, West Germany</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">aequationes mathematicae</title><title level="j" type="abbrev">Aeq. Math.</title><idno type="ISSN">0001-9054</idno><idno type="eISSN">1420-8903</idno><imprint><publisher>Birkhäuser-Verlag</publisher><pubPlace>Basel</pubPlace><date type="published" when="1989-06-01">1989-06-01</date><biblScope unit="volume">37</biblScope><biblScope unit="issue">2-3</biblScope><biblScope unit="page" from="233">233</biblScope><biblScope unit="page" to="251">251</biblScope></imprint><idno type="ISSN">0001-9054</idno></series><idno type="istex">AF5109B4D456E24567006A47CE18213D345CB6B7</idno><idno type="DOI">10.1007/BF01836446</idno><idno type="ArticleID">BF01836446</idno><idno type="ArticleID">Art8</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0001-9054</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Summary: By a well-known theorem of Lebesgue and Fréchet every measurable additive real function is continuous. This result was improved by Ostrowski who showed that a (Jensen-) convex real function must be continuous if it is bounded above on a set of positive Lebesgue measure. Recently, R. Trautner provided a short and elegant proof of the Lebesgue—Fréchet theorem based on a representation theorem for sequences on the real line. We consider here a locally compact topological groupX with some Haar measure. Then the following generalizes Trautner's theorem: Theorem.Let M be a measurable subset of X of positive finite Haar measure. Then there is a neighbourhood W of the identity e such that for each sequence (z n )in W there is a subsequence (z nk )and points y and x k in M with z nk =x k ·y −1 for k ∈ℕ. Using this theorem we obtain the following extensions of the theorems of Lebesgue and Fréchet and of Ostrowski. Theorem.Let R and T be topological spaces. Suppose that R has a countable base and that X is metrizable. If g: X → R and H: R × X → T are mappings where g is measurable on a set M of positive finite Haar measure and H is continuous in its first variable, then any solution f: X → T of f(x · y) = H(g)(x), y) for x, y∈X is continuous. Theorem.Let G: X × X → ℝ be a mapping. If there is a subset M of X of positive finite Haar measure such that for each y∈X the mapping x ↦ G(x, y) is bounded above on M, then any solution f: x → ℞ of f(x · y) ⩽ G(x, y) for x, y∈X is locally bounded above. We also prove category analogues of the above results and obtain similar results for general binary mappings in place of the group operation in the argument off.</div></front></TEI><istex><corpusName>springer</corpusName><author><json:item><name>Karl-Goswin Grosse-Erdmann</name><affiliations><json:string>Fachbereich IV—Mathematik, Universität Trier, Postfach 3825, D-5500, Trier, West Germany</json:string></affiliations></json:item></author><articleId><json:string>BF01836446</json:string><json:string>Art8</json:string></articleId><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><abstract>Summary: By a well-known theorem of Lebesgue and Fréchet every measurable additive real function is continuous. This result was improved by Ostrowski who showed that a (Jensen-) convex real function must be continuous if it is bounded above on a set of positive Lebesgue measure. Recently, R. Trautner provided a short and elegant proof of the Lebesgue—Fréchet theorem based on a representation theorem for sequences on the real line. We consider here a locally compact topological groupX with some Haar measure. Then the following generalizes Trautner's theorem: Theorem.Let M be a measurable subset of X of positive finite Haar measure. Then there is a neighbourhood W of the identity e such that for each sequence (z n )in W there is a subsequence (z nk )and points y and x k in M with z nk =x k ·y −1 for k ∈ℕ. Using this theorem we obtain the following extensions of the theorems of Lebesgue and Fréchet and of Ostrowski. Theorem.Let R and T be topological spaces. Suppose that R has a countable base and that X is metrizable. If g: X → R and H: R × X → T are mappings where g is measurable on a set M of positive finite Haar measure and H is continuous in its first variable, then any solution f: X → T of f(x · y) = H(g)(x), y) for x, y∈X is continuous. Theorem.Let G: X × X → ℝ be a mapping. If there is a subset M of X of positive finite Haar measure such that for each y∈X the mapping x ↦ G(x, y) is bounded above on M, then any solution f: x → ℞ of f(x · y) ⩽ G(x, y) for x, y∈X is locally bounded above. We also prove category analogues of the above results and obtain similar results for general binary mappings in place of the group operation in the argument off.</abstract><qualityIndicators><score>8</score><pdfVersion>1.3</pdfVersion><pdfPageSize>425 x 641 pts</pdfPageSize><refBibsNative>false</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>1677</abstractCharCount><pdfWordCount>8252</pdfWordCount><pdfCharCount>35174</pdfCharCount><pdfPageCount>19</pdfPageCount><abstractWordCount>321</abstractWordCount></qualityIndicators><title>Regularity properties of functional equations and inequalities</title><refBibs><json:item><host><author><json:item><name>Aczi~l </name></json:item><json:item><name>J </name></json:item></author><title>Lectures on functional equations and their applications</title><publicationDate>1966</publicationDate></host></json:item><json:item><author><json:item><name>Borwein </name></json:item><json:item><name>D </name></json:item><json:item><name>Ditor </name></json:item><json:item><name>S,Z </name></json:item></author><host><volume>21</volume><pages><last>498</last><first>497</first></pages><author></author><title>Canad. 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Oxford Ser</title><publicationDate>1987</publicationDate></host><title>A covering principle in real analysis</title><publicationDate>1987</publicationDate></json:item><json:item><author><json:item><name>Well </name></json:item><json:item><name>A </name></json:item></author><host><pages><last>5500</last><first>19</first></pages><author></author><title>Fachbereich IV--Mathematik</title><publicationDate>1940</publicationDate></host><title>L'intdgration clans les groupes topologiques et ses applications</title><publicationDate>1940</publicationDate></json:item></refBibs><genre><json:string>research-article</json:string></genre><host><volume>37</volume><pages><last>251</last><first>233</first></pages><issn><json:string>0001-9054</json:string></issn><issue>2-3</issue><subject><json:item><value>Analysis</value></json:item><json:item><value>Combinatorics</value></json:item></subject><journalId><json:string>10</json:string></journalId><genre><json:string>journal</json:string></genre><language><json:string>unknown</json:string></language><eissn><json:string>1420-8903</json:string></eissn><title>aequationes mathematicae</title><publicationDate>1989</publicationDate><copyrightDate>1989</copyrightDate></host><categories><wos><json:string>science</json:string><json:string>mathematics, applied</json:string><json:string>mathematics</json:string></wos><scienceMetrix><json:string>natural sciences</json:string><json:string>mathematics & statistics</json:string><json:string>general mathematics</json:string></scienceMetrix></categories><publicationDate>1989</publicationDate><copyrightDate>1989</copyrightDate><doi><json:string>10.1007/BF01836446</json:string></doi><id>AF5109B4D456E24567006A47CE18213D345CB6B7</id><score>0.63149273</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/document/AF5109B4D456E24567006A47CE18213D345CB6B7/fulltext/pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/document/AF5109B4D456E24567006A47CE18213D345CB6B7/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/AF5109B4D456E24567006A47CE18213D345CB6B7/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a" type="main" xml:lang="en">Regularity properties of functional equations and inequalities</title><respStmt><resp>Références bibliographiques récupérées via GROBID</resp><name resp="ISTEX-API">ISTEX-API (INIST-CNRS)</name></respStmt><respStmt><resp>Références bibliographiques récupérées via GROBID</resp><name resp="ISTEX-API">ISTEX-API (INIST-CNRS)</name></respStmt></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>Birkhäuser-Verlag</publisher><pubPlace>Basel</pubPlace><availability><p>Birkhäuser Verlag, 1989</p></availability><date>1987-07-24</date></publicationStmt><notesStmt><note>Research Papers</note></notesStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">Regularity properties of functional equations and inequalities</title><author xml:id="author-1"><persName><forename type="first">Karl-Goswin</forename><surname>Grosse-Erdmann</surname></persName><affiliation>Fachbereich IV—Mathematik, Universität Trier, Postfach 3825, D-5500, Trier, West Germany</affiliation></author></analytic><monogr><title level="j">aequationes mathematicae</title><title level="j" type="abbrev">Aeq. Math.</title><idno type="journal-ID">10</idno><idno type="pISSN">0001-9054</idno><idno type="eISSN">1420-8903</idno><idno type="issue-article-count">16</idno><idno type="volume-issue-count">3</idno><imprint><publisher>Birkhäuser-Verlag</publisher><pubPlace>Basel</pubPlace><date type="published" when="1989-06-01"></date><biblScope unit="volume">37</biblScope><biblScope unit="issue">2-3</biblScope><biblScope unit="page" from="233">233</biblScope><biblScope unit="page" to="251">251</biblScope></imprint></monogr><idno type="istex">AF5109B4D456E24567006A47CE18213D345CB6B7</idno><idno type="DOI">10.1007/BF01836446</idno><idno type="ArticleID">BF01836446</idno><idno type="ArticleID">Art8</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>1987-07-24</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>Summary: By a well-known theorem of Lebesgue and Fréchet every measurable additive real function is continuous. This result was improved by Ostrowski who showed that a (Jensen-) convex real function must be continuous if it is bounded above on a set of positive Lebesgue measure. Recently, R. Trautner provided a short and elegant proof of the Lebesgue—Fréchet theorem based on a representation theorem for sequences on the real line. We consider here a locally compact topological groupX with some Haar measure. Then the following generalizes Trautner's theorem: Theorem.Let M be a measurable subset of X of positive finite Haar measure. Then there is a neighbourhood W of the identity e such that for each sequence (z n )in W there is a subsequence (z nk )and points y and x k in M with z nk =x k ·y −1 for k ∈ℕ. Using this theorem we obtain the following extensions of the theorems of Lebesgue and Fréchet and of Ostrowski. Theorem.Let R and T be topological spaces. Suppose that R has a countable base and that X is metrizable. If g: X → R and H: R × X → T are mappings where g is measurable on a set M of positive finite Haar measure and H is continuous in its first variable, then any solution f: X → T of f(x · y) = H(g)(x), y) for x, y∈X is continuous. Theorem.Let G: X × X → ℝ be a mapping. If there is a subset M of X of positive finite Haar measure such that for each y∈X the mapping x ↦ G(x, y) is bounded above on M, then any solution f: x → ℞ of f(x · y) ⩽ G(x, y) for x, y∈X is locally bounded above. We also prove category analogues of the above results and obtain similar results for general binary mappings in place of the group operation in the argument off.</p></abstract><textClass><keywords scheme="Journal Subject"><list><head>Mathematics</head><item><term>Analysis</term></item><item><term>Combinatorics</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1987-07-24">Created</change><change when="1989-06-01">Published</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2016-11-22">References added</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2017-01-20">References added</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/AF5109B4D456E24567006A47CE18213D345CB6B7/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="Springer, Publisher found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="UTF-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//Springer-Verlag//DTD A++ V2.4//EN" URI="http://devel.springer.de/A++/V2.4/DTD/A++V2.4.dtd" name="istex:docType"></istex:docType><istex:document><Publisher><PublisherInfo><PublisherName>Birkhäuser-Verlag</PublisherName><PublisherLocation>Basel</PublisherLocation></PublisherInfo><Journal><JournalInfo JournalProductType="ArchiveJournal" NumberingStyle="Unnumbered"><JournalID>10</JournalID><JournalPrintISSN>0001-9054</JournalPrintISSN><JournalElectronicISSN>1420-8903</JournalElectronicISSN><JournalTitle>aequationes mathematicae</JournalTitle><JournalAbbreviatedTitle>Aeq. Math.</JournalAbbreviatedTitle><JournalSubjectGroup><JournalSubject Type="Primary">Mathematics</JournalSubject><JournalSubject Type="Secondary">Analysis</JournalSubject><JournalSubject Type="Secondary">Combinatorics</JournalSubject></JournalSubjectGroup></JournalInfo><Volume><VolumeInfo VolumeType="Regular" TocLevels="0"><VolumeIDStart>37</VolumeIDStart><VolumeIDEnd>37</VolumeIDEnd><VolumeIssueCount>3</VolumeIssueCount></VolumeInfo><Issue IssueType="Combined"><IssueInfo TocLevels="0"><IssueIDStart>2</IssueIDStart><IssueIDEnd>3</IssueIDEnd><IssueArticleCount>16</IssueArticleCount><IssueHistory><CoverDate><DateString>1989</DateString><Year>1989</Year><Month>6</Month></CoverDate></IssueHistory><IssueCopyright><CopyrightHolderName>Birkhäuser Verlag</CopyrightHolderName><CopyrightYear>1989</CopyrightYear></IssueCopyright></IssueInfo><Article ID="Art8"><ArticleInfo Language="En" ArticleType="OriginalPaper" NumberingStyle="Unnumbered" TocLevels="0" ContainsESM="No"><ArticleID>BF01836446</ArticleID><ArticleDOI>10.1007/BF01836446</ArticleDOI><ArticleSequenceNumber>8</ArticleSequenceNumber><ArticleTitle Language="En">Regularity properties of functional equations and inequalities</ArticleTitle><ArticleCategory>Research Papers</ArticleCategory><ArticleFirstPage>233</ArticleFirstPage><ArticleLastPage>251</ArticleLastPage><ArticleHistory><RegistrationDate><Year>2005</Year><Month>5</Month><Day>20</Day></RegistrationDate><Received><Year>1987</Year><Month>7</Month><Day>24</Day></Received><Accepted><Year>1989</Year><Month>1</Month><Day>6</Day></Accepted></ArticleHistory><ArticleCopyright><CopyrightHolderName>Birkhäuser Verlag</CopyrightHolderName><CopyrightYear>1989</CopyrightYear></ArticleCopyright><ArticleGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ArticleGrants><ArticleContext><JournalID>10</JournalID><VolumeIDStart>37</VolumeIDStart><VolumeIDEnd>37</VolumeIDEnd><IssueIDStart>2</IssueIDStart><IssueIDEnd>3</IssueIDEnd></ArticleContext></ArticleInfo><ArticleHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>Karl-Goswin</GivenName><FamilyName>Grosse-Erdmann</FamilyName></AuthorName></Author><Affiliation ID="Aff1"><OrgDivision>Fachbereich IV—Mathematik</OrgDivision><OrgName>Universität Trier</OrgName><OrgAddress><Postbox>Postfach 3825</Postbox><Postcode>D-5500</Postcode><City>Trier</City><Country>West Germany</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Summary</Heading><Para>By a well-known theorem of Lebesgue and Fréchet every measurable additive real function is continuous. This result was improved by Ostrowski who showed that a (Jensen-) convex real function must be continuous if it is bounded above on a set of positive Lebesgue measure. Recently, R. Trautner provided a short and elegant proof of the Lebesgue—Fréchet theorem based on a representation theorem for sequences on the real line.</Para><Para>We consider here a locally compact topological group<Emphasis Type="Italic">X</Emphasis> with some Haar measure. Then the following generalizes Trautner's theorem:</Para><Para><Emphasis Type="SmallCaps">Theorem</Emphasis>.<Emphasis Type="Italic">Let M be a measurable subset of X of positive finite Haar measure. Then there is a neighbourhood W of the identity e such that for each sequence (z</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>)<Emphasis Type="Italic">in W there is a subsequence (z</Emphasis><Subscript><Emphasis Type="Italic">nk</Emphasis></Subscript>)<Emphasis Type="Italic">and points y and x</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript><Emphasis Type="Italic">in M with z</Emphasis><Subscript><Emphasis Type="Italic">nk</Emphasis></Subscript> =<Emphasis Type="Italic">x</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript> ·<Emphasis Type="Italic">y</Emphasis><Superscript>−1</Superscript><Emphasis Type="Italic">for k ∈ℕ.</Emphasis></Para><Para>Using this theorem we obtain the following extensions of the theorems of Lebesgue and Fréchet and of Ostrowski.</Para><Para><Emphasis Type="SmallCaps">Theorem</Emphasis>.<Emphasis Type="Italic">Let R and T be topological spaces. Suppose that R has a countable base and that X is metrizable. If g: X → R and H: R × X → T are mappings where g is measurable on a set M of positive finite Haar measure and H is continuous in its first variable, then any solution f: X → T of f(x · y) = H(g)(x), y) for x, y∈X is continuous.</Emphasis></Para><Para><Emphasis Type="SmallCaps">Theorem</Emphasis>.<Emphasis Type="Italic">Let G: X × X → ℝ be a mapping. If there is a subset M of X of positive finite Haar measure such that for each y∈X the mapping x ↦ G(x, y) is bounded above on M, then any solution f: x → ℞ of f(x · y) ⩽ G(x, y) for x, y∈X is locally bounded above.</Emphasis></Para><Para>We also prove category analogues of the above results and obtain similar results for general binary mappings in place of the group operation in the argument of<Emphasis Type="Italic">f.</Emphasis></Para></Abstract><KeywordGroup Language="En"><Heading>AMS (1980) subject classification</Heading><Keyword>Primary 39B70</Keyword><Keyword>Secondary 39C05</Keyword></KeywordGroup></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Regularity properties of functional equations and inequalities</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Regularity properties of functional equations and inequalities</title></titleInfo><name type="personal"><namePart type="given">Karl-Goswin</namePart><namePart type="family">Grosse-Erdmann</namePart><affiliation>Fachbereich IV—Mathematik, Universität Trier, Postfach 3825, D-5500, Trier, West Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper"></genre><originInfo><publisher>Birkhäuser-Verlag</publisher><place><placeTerm type="text">Basel</placeTerm></place><dateCreated encoding="w3cdtf">1987-07-24</dateCreated><dateIssued encoding="w3cdtf">1989-06-01</dateIssued><copyrightDate encoding="w3cdtf">1989</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">Summary: By a well-known theorem of Lebesgue and Fréchet every measurable additive real function is continuous. This result was improved by Ostrowski who showed that a (Jensen-) convex real function must be continuous if it is bounded above on a set of positive Lebesgue measure. Recently, R. Trautner provided a short and elegant proof of the Lebesgue—Fréchet theorem based on a representation theorem for sequences on the real line. We consider here a locally compact topological groupX with some Haar measure. Then the following generalizes Trautner's theorem: Theorem.Let M be a measurable subset of X of positive finite Haar measure. Then there is a neighbourhood W of the identity e such that for each sequence (z n )in W there is a subsequence (z nk )and points y and x k in M with z nk =x k ·y −1 for k ∈ℕ. Using this theorem we obtain the following extensions of the theorems of Lebesgue and Fréchet and of Ostrowski. Theorem.Let R and T be topological spaces. Suppose that R has a countable base and that X is metrizable. If g: X → R and H: R × X → T are mappings where g is measurable on a set M of positive finite Haar measure and H is continuous in its first variable, then any solution f: X → T of f(x · y) = H(g)(x), y) for x, y∈X is continuous. Theorem.Let G: X × X → ℝ be a mapping. If there is a subset M of X of positive finite Haar measure such that for each y∈X the mapping x ↦ G(x, y) is bounded above on M, then any solution f: x → ℞ of f(x · y) ⩽ G(x, y) for x, y∈X is locally bounded above. We also prove category analogues of the above results and obtain similar results for general binary mappings in place of the group operation in the argument off.</abstract><note>Research Papers</note><relatedItem type="host"><titleInfo><title>aequationes mathematicae</title></titleInfo><titleInfo type="abbreviated"><title>Aeq. Math.</title></titleInfo><genre type="journal" displayLabel="Archive Journal"></genre><originInfo><dateIssued encoding="w3cdtf">1989-06-01</dateIssued><copyrightDate encoding="w3cdtf">1989</copyrightDate></originInfo><subject><genre>Mathematics</genre><topic>Analysis</topic><topic>Combinatorics</topic></subject><identifier type="ISSN">0001-9054</identifier><identifier type="eISSN">1420-8903</identifier><identifier type="JournalID">10</identifier><identifier type="IssueArticleCount">16</identifier><identifier type="VolumeIssueCount">3</identifier><part><date>1989</date><detail type="volume"><number>37</number><caption>vol.</caption></detail><detail type="issue"><number>2-3</number><caption>no.</caption></detail><extent unit="pages"><start>233</start><end>251</end></extent></part><recordInfo><recordOrigin>Birkhäuser Verlag, 1989</recordOrigin></recordInfo></relatedItem><identifier type="istex">AF5109B4D456E24567006A47CE18213D345CB6B7</identifier><identifier type="DOI">10.1007/BF01836446</identifier><identifier type="ArticleID">BF01836446</identifier><identifier type="ArticleID">Art8</identifier><accessCondition type="use and reproduction" contentType="copyright">Birkhäuser Verlag, 1989</accessCondition><recordInfo><recordContentSource>SPRINGER</recordContentSource><recordOrigin>Birkhäuser Verlag, 1989</recordOrigin></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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