Then

IMSLectureNotes{MonographSeriesVol.0(2005)1{4NearlysimultaneousproofsoftheErgodicTheoremandMaximalErgodicTheoremMichaelKeaneandKarlPetersenWesleyanUniversityandUniversityofNorthCarolinaAbstract:WegiveashortproofofastrengtheningoftheMaximalErgodicTheoremwhichalsoimmediatelyyieldsthePointwiseErgodicTheorem.Let(X;B;)beaprobabilityspace,T:X!Xa(possiblynoninvertible)measure-preservingtransformation,andf2L1(X;B;).LetAkf=1kk1Xj=0fTj;fN=sup1kNAkf;f=supNfN;andA=limsupk!1Akf:Whenisaconstant,thefollowingresultistheMaximalErgodicTheorem.Choos-ing=AcoversmostoftheproofoftheErgodicTheorem.Theorem.Letbeaninvariant(T=a.e.)functiononXwith+2L1.ThenZffg(f)0:Proof.Wemayassumethat2L1ffg,sinceotherwiseZffg(f)=10:Butthenactually2L1(X),sinceonffgwehavef,sothatonthissetf++,whichisintegrable.Assumerstthatf2L1.FixN=1;2;:::,andletEN=ffNg:Noticethat(f)EN(f);sincex=2ENimplies(f)(x)0.ThusforaverylargemN,wecanbreakupm1Xk=0(f)EN(Tkx)intoconvenientstringsoftermsasfollows.Thereismaybeaninitialstringof0'sduringwhichTkx=2EN.ThenthereisarsttimekwhenTkx2EN,whichinitiatesastringofnomorethanNterms,thesumofwhichispositive(usingonAMS2000subjectclassications:Primary37A30;secondary37A05Keywordsandphrases:Maximalergodictheorem,pointwiseergodictheorem1imsart-lnmsver.2005/02/28file:KeanePetersenLNMS.texdate:March10,2005M.Keane,K.Petersen/ProofsoftheErgodicTheoremandMaximalErgodicTheorem2eachofthesetermsthefactthat(f)EN(f)).Beginningafterthelastterminthisstring,werepeatthepreviousanalysis,ndingmaybesome0'suntilagainsomeTkx2ENinitiatesanotherstringofnomorethanNtermsandwithpositivesum.Thefullsumofmtermsmayendinthemiddleofeitherofthesetwokindsofstrings(0's,orhavingpositivesum).Thuswecanndj=mN+1;:::;msuchthatm1Xk=0(f)EN(Tkx)m1Xk=j(f)EN(Tkx)N(kfk1++(x)):Integratingbothsides,dividingbym,andlettingm!1givesmZEN(f)N(kfk1+k+k1);ZEN(f)Nm(kfk1+k+k1);ZEN(f)0:LettingN!1andusingtheDominatedConvergenceTheoremconcludestheproofforthecasef2L1.Toextendtothecasef2L1,fors=1;2;:::lets=f
fjfjsg,sothats2L1ands!fa.e.andinL1.ThenforxedN(s)N!fNa.e.andinL1and(f(s)Ng4ffNg)!0:Therefore0Zf(s)Ng(s)!ZffNg(f);againbytheDominatedConvergenceTheorem.ThefullresultfollowsbylettingN!1.Corollary(ErgodicTheorem).Thesequence(Akf)convergesa.e..Proof.ItisenoughtoshowthatZAZf:Forthen,lettingA=liminfAkf,applyingthistofgivesZAZf;sothatZAZfZAZA;andhenceZ(AA)=0;sothatA=Aa.e..Considerrstf+anditsassociatedA,denotedbyA(f+).ForanyinvariantfunctionA(f+)suchthat+2L1,forexample=A(f+)^n1=n,wehavef(f+)g=X,sotheTheoremgivesZf+Z%ZA(f+):imsart-lnmsver.2005/02/28file:KeanePetersenLNMS.texdate:March10,2005M.Keane,K.Petersen/ProofsoftheErgodicTheoremandMaximalErgodicTheorem3Thus(A)+A(f+)isintegrable(and,byasimilarargument,sois(A)A(f).)Nowlet0bearbitraryandapplytheTheoremto=AtoconcludethatZfZ%ZA:Remark.ThisproofmayberegardedasafurtherdevelopmentofonegiveninapaperbyKeane[10],whichhasbeenextendedtodealalsowiththeHopfRa-tioErgodicTheorem[8]andwiththecaseofhigher-dimensionalactions[11],andwhichwasitselfadevelopmentoftheKatznelson-Weissproof[9]basedonKamae'snonstandard-analysisproof[7].(ItispresentedalsointheBedford-Keane-Seriescol-lection[1].)OurproofyieldsboththePointwiseandMaximalErgodicTheoremses-sentiallysimultaneouslywithoutaddinganyrealcomplications.Roughlycontempo-raneouslywiththisformulation,RolandZweimullerpreparedsomepreprints[21,22]alsogivingshortproofsbasedontheKamae-Katznelson-Weissapproach,andre-centlyhehasalsoproducedasimpleproofoftheHopftheorem[23].WithoutgoingtoodeepintothecomplicatedhistoryoftheErgodicTheoremandMaximalErgodicTheorem,itisinterestingtonotesomerecurrencesastheuseofmaximaltheoremsaroseandwanedrepeatedly.AftertheoriginalproofsbyvonNeumann[18],Birkho[2],andKhinchine[12],theroleandimportanceoftheMaximalLemmaandMax-imalTheoremwerebroughtoutbyWiener[19]andYosida-Kakutani[20],makingpossibletheexplorationofconnectionswithharmonicfunctionsandmartingales.Proofsbyupcrossingsfollowedananalogouspattern.Italsobecameofinterest,forinstancetoallowextensiontonewareasornewkindsofaverages,againtoprovetheErgodicTheoremwithoutresorttomaximallemmasortheorems,asintheproofbyShields[16]inspiredbytheOrnstein-WeissproofoftheShannon-McMillan-BreimanTheoremforactionsofamenablegroups[14],orinBourgain'sproofsbymeansofvariationalinequalities[3].Sometimesitwaspointedout,forexampleinthenotebyR.Jones[6],thattheseapproachescouldalsowithveryslightmodicationprovetheMaximalErgodicTheorem.OfcoursetherearethetheoremsofStein[17]andSawyer[15]thatmaketheconnectionexplicit,justasthetransferencetechniquesofWiener[19]andCalderon[4]connectergodictheoremswiththeiranaloguesinanalysisliketheHardy-LittlewoodMaximalLemma[5].Inmanyoftheimprovementsovertheyears,ideasandtricksalreadyinthepapersofBirkho,Kolmogorov[13],Wiener,andYosida-Kakutanihavecontinuedtoplayanessentialrole.AcknowledgmentThisnotearoseoutofaconversationbetweentheauthorsin1997attheErwinSchrodingerInternationalInstitutefo