Foundations of Regular Variation by

FoundationsofRegularVariationbyN.H.BinghamandA.J.Ostaszewski(London)InmemoriamPaulErd}os,1913-1996Abstract.Thetheoryofregularvariationislargelycompleteinonedimension,butisdevelopedunderregularityorsmoothnessassumptions.Forfunctionsofarealvariable,Lebesguemeasurabilitysu±ces,andsodoeshavingtheprop-ertyofBaire.We¯ndherethattheprecedingtwopropertieshavetwokindsofcommongeneralization,bothofacombinatorialnature;oneisexempli¯edby`containmentuptotranslationofsubsequences',theother,drawnfromde-scriptivesettheory,requiresnon-emptinessofaSouslin¢12-set.Allofourgeneralizationsareequivalenttotheuniformconvergenceproperty.1Introductionandresults1.1PreambleThetheoryofregularvariation,orofregularlyvaryingfunctions,isachapterintheclassicaltheoryoffunctionsofarealvariable,datingfromtheworkofKaramatain1930.Ithasfoundextensiveuseinprobabilitytheory,analysis(particularlyTauberiantheoryandcomplexanalysis),numbertheoryandotherareas;see[BGT]foramonographtreatment,and[Kor]ChapterIV.Henceforthweidentifyournumerousreferencesto[BGT]byBGT.Thetheoryexplorestheconsequencesofarelationshipoftheformf(¸x)=f(x)!g(¸)(x!1)8¸0;(RV)forfunctionsde¯nedonR+:ThelimitfunctiongmustsatisfytheCauchyfunc-tionalequationg(¸¹)=g(¸)g(¹)8¸;¹0:(CFE)Subjecttoamildregularitycondition,(CFE)forcesgtobeapower:g(¸)=¸½8½0:(½)Thenfissaidtoberegularlyvaryingwithindex½,writtenf2R½.AMSSubjectClassi¯cation:26AO3;04A15;02K20.Keywords:Regularvari-ation,measurability,Baireproperty,uniformconvergencetheorem,Karamatatheory,Cauchyfunctionalequation,Hamelpathology,descriptivesettheory,axiomofdeterminacy,combinatorialprinciples`club'(|)andNoTrumps,au-tomaticcontinuity,similarsequence.1Thecase½=0isbasic.Afunctionf2R0iscalledslowlyvarying;slowlyvaryingfunctionsareoftenwritten`(forlente,orlangsam).ThebasictheoremofthesubjectistheUniformConvergenceTheorem(UCT),whichstatesthatif`(¸x)=`(x)!1(x!1)8¸0;(SV)thentheconvergenceisuniformoncompact¸-setsin(0;1).Thebasicfactsare:(i)if`is(Lebesgue)measurable,thentheUCTholds;(ii)if`hastheBaireproperty(forwhichseee.g.Kuratowski[Kur],Oxtoby[Oxt]),thentheUCTholds;(iii)ingeneral,theUCTneednothold.Similarly,iffismeasurableorhastheBaireproperty,(CFE)implies(½),butnotingeneral.SeeBGTxx1.1,1.2;forbackgroundontheCauchyfunctionalequation,see[Kucz],[AD].Althoughinthiscontextmeasureandcategoryareinterchange-able,wewillwarnthereaderinSection5thatinterchangeabilityisnotguaran-teed.TheUCTextendseasilytoregularlyaswellasslowlyvaryingfunctions;seeBGTTh.1.5.2.Thebasiccaseis½=0,sowelosenothingbyrestrictingattentiontoithere.Thebasicfoundationalquestioninthesubject,whichweaddresshere,con-cernsthesearchfornaturalconditionsfortheabovetohold,andinparticularforasubstantialcommongeneralizationofmeasurabilityandtheBaireprop-erty.We¯ndsuchacommongeneralization,indeedtwokindsofgeneralization,whichareactuallybothnecessaryandsu±cient.ThepaperthusanswersanoldproblemnotedinBGTp.11Section1.2.5.Whileregularvariationisusuallyusedinthemultiplicativeformulationabove,forproofsinthesubjectitisusuallymoreconvenienttouseanad-ditiveformulation.Writingh(x):=logf(ex)(orlog`(ex)asthecasemaybe),k(u):=logg(eu),therelationsabovebecomeh(x+u)¡h(x)!k(u)(x!1)8u2R;(RV+)h(x+u)¡h(x)!0(x!1)8u2R;(SV+)k(u+v)=k(u)+k(v)8u;v2R:(CFE+)Herethefunctionsarede¯nedonR;whereasinthemultiplicativenotationfunctionsarede¯nedonR+:InBGT,conditionsareimposedonfunctionsf.Itismorehelpfulheretoidentifyafunctionwithitsgraph{sothaty=f(x)means(x;y)2f,etc.Conditionsarethusimposedonsets,andweareabletousethelanguageofdescriptivesettheory,forwhichseee.g.[Mos],asinx2below.SeethecommentinSection5onclassi¯cationbyreferencetopre-images(whichneedsthematerialofSection2).2ItisconvenienttodescribethecontextoftheUniformConvergenceTheorem(UCT)bywritinghx(u)=h(u+x)¡h(x)andregardinghx(u);withxasparameter,asthoughitwerean`approximately-additive'functionofu(atermde¯nedexplicitlyin[Kucz]p.424).Then,grantedassumptionsonthefunctionh;(UCT)assertsthatpointwiseconver-genceofthefamilyfhxgimpliesuniformconvergenceovercompactsetsofu.Theentireanalysisrestsontwokeyde¯nitionsandonepurelyset-theoreticcombinatorialprinciplethatcanaddresspracticalitieswithin`naivesettheory'(withoutanyneedforformalaxiomatics).InSection5westate,butdonotprove,anewtheoreminthestyleofHeiberg'sTheoremaimedattestingforUCTbyemployingthoseofourresultsthatrelyonlyon`naivesettheory'.Foraproofofthisandofmorepowerfulvariantsseethecompanionpaper[BOst2].Therelevantcombinatorialprincipleanditsvariantsarede¯nedinSection1.2andusedtoestablishanumberofinterestingnececessaryandsu±cientconditionsequivalenttoUCT.InSection1.3we¯ndafurthersetofnecessaryandsu±cientconditionsequivalenttoUCTwhichrequiresomeknowledgeof`de¯nability'asformalizedbyDescriptiveSetTheory.Thepointofviewadoptedthereleadstoconnectionswithin¯nitarygames(analternative`canon'forcombinatorialprinciples,callingforfurtherresearchintoUCT).Thatsubsectionhasbeenwrittensothatitcanbereadwithaminimalexposuretotechnicalitiesandsowerecommendthatanalystswhoprefertodependonlyon`naivesettheory'shouldnot°inchfromreadingthis.Indeed,toplacethemint