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1、Hausdorff微积分及其若干应用Northeast.MathJ.14(4)(1998),479—488(4钾,一HausdorffCalcul圻fApplications'Dt]7(Library.TeachersCollege.JimeiUniversitytXiamen,F~.ii361021)AbstractTheconceptsofHausdorffderivativeandHausdorffintegralwereintroducedbyS.S.FuFurther,wefindthataLebesguesingularfunctioncanusua
2、llybedecomposedintothesumofanabsolutelycontinuousfunctionandasingularfunctioninHausdorffsense.Thereforeweobtainamoregenera[theoremofLebesguedecomposition.AndinmanycasestaLebesguesingularfunctionisJustanabsolutelycontinuousfunctioninHausdorffsensetsowehaveanintegralrepresentation.Asit
3、sapplication,abetterformularofintegralexpressionofthelengthofanarcisg1veil.ThefollowingconceptsandtheirrelatedpropertieslistedinthissectioncanbefoundinI-t,2,3].1.Anon—negativefunction(E)definedonallsetsEofametricspaceMiscalledameasureifitfulfillsthefollowingconditions:(m1)()一0;(m2)(u
4、E.)≤∑(E)foreachsequence{E.}ofsets..JAmeasureonMisametricifitfulfillsthecondition(ma)(E】UE2)一(E1)_卜(E2)wheneverdist(E1,E2)>O.AsetECMissaidtobe—measurableormeasurablewithrespecttothemeasureif(A)一(AnE)+/t(A'~E)foreachsetA[MRemarkAssometextsdid(see[4]),wedonotdistinguishmeasureandoute
5、rmeasure.AmeasureonMisregularifforeachsetACMthereexistsa—measurablesetEsuchthatA[Eand(A)一(E).LetbearegularmeasureonM.If{AK}isanincreasingsequenceofsets,thenReceivedOct7.1997*)Pro~ectpardysupportedbyJUTCSF伊童翼陆(;l1l辅～s漱hN0RTHEAST.MATH.JVOL】41im/t(AK)一(UA).WeremarkthatthesetsAneednotbe一
6、measurablehere.2.Inthispaper,thespaceMis,,aclosedbounedintervalontherealline.SupposeE[,6].Asequenceofopenintervals{I.}iscalledacoverofEifE[UI.,andiscalleda一coverofEif,further,0<I,.1≤foreach,wherelJlisthelengthof.LetE[[日,6]and0<≤1.For>0,define电(E)一inf∑lJlwheretheinfimumistake
7、noverall一covers{)ofE.TheHausdorffsdimensionalmeasureofEisdefinedby(E)一lim(E).when—l,wewrite:,i.e.,istheLebesguemeasure.Itiswellknownthat矽isaregularmetricmeasure.3.AcollectionofsetsiscalledaVitaliclassforEifforeach∈Eandd>0thereexistsU∈with∈Uand0<lul≤,wherel【,listhediameterofU.Vl
8、tallCovering
