符号测度中Hahn分解定理的推广

:16730291(2008)03008707Hahn刘晓鹏,刘坤会(,100044):首先讨论了变差函数的上变差和下变差函数所生成的测度及符号测度Hahn分解中的具体问题,在此基础上得到了变差过程生成测度Jordan分解的一个结论.此外对经典Hahn分解Radon导数及条件期望的概念作了进一步的推广并得到了相应的结论.:变差函数;变差过程;测度;Hahn分解;条件期望:O211.6:AExtensionofHahnResolutionTheoremonSymbolicMeasureLIUXiaopeng,LIUKunhui(SchoolofScience,BeijingJiaotongUniversity,Beijing100044,China)Abstract:ThepaperdiscussessomeconcreteproblemsaboutgeneratedmeasureofuppervariationandlowervariationfunctionofvariationalfunctionandHahnresolutionofsymbolicmeasureatfirst.ItgetsaconclusionaboutJordanresolutionofgeneratedmeasureofvariationalprocessonthebasisoftheformerconclusions.Besides,thispaperextendstheclassicalconceptsofHahnresolution,Radonderivativeandconditionalexpectationfurther,andobtainstherelevantconclusions.Keywords:variationalfunction;variationalprocess;measure;Hahnresolution;conditionalexpectation:20061128:(1980!),,,.email:[e-mail](1947!),,,,,.1Hahn.,.,,[110].,.,,.Hahn,HahnRadon,.(,F,P),(Ft),t∀0F-.At,t∀0(,F,P),A0-=0.AA..t#R+#,At()=|A0()|+∃t0(A()),∃t0(A())A!()[0,t][1112],At()A.()[0-,t].A+=A+A2,A-=A-A2,A+,A-A+-A-=A,A,A=A+-A-A.A,B(R+)%F!A(H)=E[&[0,∋)IH(s,(),dAs(()],32320086JOURNALOFBEIJINGJIAOTONGUNIVERSITYVol.32No.3Jun.2008H#B(R+)%F(1)!AB(R+)%F,A.[1314].A,HB(R+)%F!A+(H)!A-(H),!A∀!A+-!A-G∀H)(B(R+)%F),AH.Hahn[15],!A,HahnJordan?:1A(,F,P),R+%#H#B(R+)%F,!AAH,!+A=!A+,!-A=!A-,!A=!A(2)(2)!A+,!A-!AG∀H)(B(R+)%F).A++A-=A,!A++!A-=!A,!A+∃!A,!A-∃!A(∃).Radon[15]B(R+)%FL(1),L(2)!A+(G)=&GL(1)d!A,!A-(G)=&GL(2)d!A,G#B(R+)%F(3)[13](3)A+=L(1)(A,A-=L(2)(A(4)(4)L(1)(AL(2)(AL(1)L(2)AStieltjes,.f=L(1)-L(2),fB(R+)%FA=A+-A-=f!A(5)(5)N#F,P(N)=0,#NcAt()=&[0,t]f(s,)dAs(),t#R+(6)B={f∗0},B#B(R+)%F.[15],#,f((,)B(R+),B()#B(R+),B()B,G#B(R+)%F,!A+(G)=!A(G)Bc)!A-(G)=!A(G)B)(7),G()#B(R+)#,(6)[16]7!A+(G)=E[&[0,∋)IG(s,()dA+s(()]=E[!+(G((),()]=E[%!(G(())Bc((),()]=E[&[0,∋)IG)Bc(s,()dAs(()]=!A(G)Bc)(8)(8)!+(C,),C#B(R+),#R+A+(()(R+,B(R+))L-S.!-(C,)%!(C,)A-((C,)A!()L-S.(8)!A-(G)=!A(G)B)(9)(8)(9)(7).G#G,G~#G,G~&G,(7)!A(G~)=!A+(G~)-!A-(G~)=!A(G~)Bc)-!A(D~)B)∗!A(G)Bc)=!A(G)Bc)=!A+(G)(10)(10)!A+=supG~&G,G~#G!A(G~),!A+(G)=!+A(G),G#G(11)!A-(G)=!-A(G),G#G(12)(11)(12)(7)!A(G)=!+A(G)+!-A(G)=!A(G),G#G(13)(11~13)(2),!A=!A+-!A-Jordan..2HahnHahn,.2!1,!2(,F)!n#F,n+n∀1!2(n)∋,!2F-,(,F)!+,!-D#F!1(A)∀!+(A)=%!(A)Dc),!2(A)∀!-(A)=%!(A)D),A#F(14)(14)%!(,F)%!=!++!-.,A#F,!1(A)!2(A)∋,!∀!1-!2(A,A)F),!+!-%!(A,A)F),!+-!-(A,A)F)!+-!-=!1-!2=!,!=!+-!-Jordan,A=A)Dc+A)DHahn8832,%!=!++!-.,!+!-(D).n∀1,!2(n)∋,!1,!2(n,n)F),∀n∀!1-!2(n,n)F).JordanHahn∀n=∀+n-∀-n,n=n)Dcn+Dn,Dn#n)F(15)A#n)F,∀+n(A)=supB&A,B#F∀n(B)=∀n(A)Dcn)(16)∀-n(A)=-infB&A,B#F∀n(B)=-∀n(A)Dn)(17)D(n)=,nj=1Dj,D(n)#n)F,A#F∀-n(A)n)=-infB&A)n,B#F∀n(B)=-∀n(A)D(n))(18)(17)A#F∀-n(A)n)=-infB&A)n,B#F∀n(B)∀-∀n(A)D(n))(19)∀-n(A)n)=−nj=1∀-n[A)(j-j-1)]=−nj=1-infB&A)(j-j-1),B#F∀n(B)=−nj=1-infB&A)(j-j-1),B#F∀j(B)=−nj=1∀-j[A)(j-j-1)]∗−nj=1∀-j[A)(j-D(j-1))]=-−nj=1∀j[A)(Dj-D(j-1))]=-−nj=1∀n[A)(Dj-D(j-1))]=-∀n(A)D(n))(20)(20)0=D(0)=#.(19)(20)(18).A#F,n∀1,∀-n(A)n)∗∀-n+1(A)n+1)(21),(18)∀-n(A)n)=-infB&A)n,B#F∀n(B)=-infB&A)n,B#F∀n+1(B)∗-infB&A)n+1,B#F∀n+1(B)=∀-n+1(A)n+1)!(21).!-(A)=limn.∋∀-n(A)n),A#F(22)!-(,F),.Aj,j∀1F,A=−∋j=1Aj,(21)(22)!-(A)∀∀-k(A)k)=−∋j=1∀-k(Aj)k),k∀1!k!-(A)∀−∋j=1!-(Aj)(23)(21)(22)k∀1∀-k(A)k)=−∋j=1∀-k(Aj)k)∗−∋j=1!-(Aj)!(22)!-(A)∗−∋j=1!-(Aj)(24)(23,24)!-(,F),!-(A)=−∋j=1!-(Aj)(25)!2(n)∋,A#Fn∀1∀n(A)n)=!1(A)n)-!2(A)n)-∋,(18)-∀n(A)n)D(n))=∀-n(A)n)∀0,(15)∀+n(A)n)=∀n(A)n)+∀-n(A)n)=∀n(A)n)-∀n(A)n)D(n))=∀n[A)n)(D(n))c](26)(16)(26),A#F,n∀1,∀+n(A)n)=supB&A)n,B#F∀n(B)=∀n[A)n)(D(n))c](27)(21)∀+n(A)n)A#Fn,!+(A)=limn.∋∀+n(A)n),A#F(28)(25)!+(,F),.(27)A#Fk∀1,∀+n(A)D(k))n)=∀n(#)=0,n∀k(29)(28)(29)A#F,!+(A)D(k))=limn.∋∀+n(A)D(k))n)=limn∀k,n.∋∀+n(A)D(k))n)=0,k∀1(30)893:HahnD=,∋j=1Dj,D#FD(k)+D,(30),A#F,!+(A)D)=limk.∋!+(A)D(k))=0(31)(18),A#Fn∀1,∀-n(A)D(n))n)=∀-n(A)n)(32)(22)(32)A#F!-(A)=limn.∋∀-n(A)n)=limn.∋∀-n(A)D(n))n)∗limn.∋∀-n(A)D)n)=!-(A)D)∗!-(A)(33)(31)(33)A#F,!+(A)Dc)=!+(A)!-(A)D)=!-(A)(34)%!=!++!-,%!(,F)A#F,!+(A)=%!(A)Dc),!-(A)=%!(A)D).A#F,k∀1,n∀k∀+n(A)k)=supB&A)k,B#F∀n(B)=supB&A)k,B#F∀k(B)=∀+k(A)k)!∀-n(A)k)=-infB&A)k,B#F∀n(B)=-infB&A)k,B#F∀k(B)=∀-k(A)k)!A#Fk∀1!+(A)k)=∀+n(A)k),n∀k(35)!-(A)k)=∀-n(A)k),n∀k(36)(16)(35)A#F!+(A)=!+(A)1)+!+[A)(2-1)]+/+!+[A)(n-n-1)]+/=∀+1(A)1)+∀+2[A)(2-1)]+/+∀+n[A)(n-n-1)]+/∗!1(A)1)+!1[A)(2-1)]+/+!1[A)(n-n-1)]+/=!1(A)!,(17)(36)A#F!-(A)∗!2(A).(14).A#F,!1(A)∋!2(A)∋.!1,!2(A,A)F),!∀!1-!2(A,A)F).!+,!-(A,A)F),!1(A)∋,B#A)F!+(B)∗!1(B)∗!1(A)∋.!2(A)∋,B#A)F!-(B)∋.!+-!-(A,A)F)!B#A)F,(15)(35)(36)!(B)=!(B)1)+![B)(2-1)]+/+![B)(n-n-1)]+/=∀1(B)1)+∀2[B)(2-1)]+/+∀n[B)(n-n-1)]+/={∀+1(B)1)-∀-1(B)1)}+{∀+2[B)(2-1)]-∀-2[B)(2-1)]}+/+{∀+n[B)(n-n-1)]-∀-n[B)(n-n-1)]}+/={!+(B)1)-!-(B)1)}+{!+[B)(2-1)]-!-[B)(2-1)]}+/+{!+[B)(n-n-1)]-!-[B)(n-n-1)]}+/=!+(B)-!-(B)(37)A#F,!1(A)∋!2(A)∋,(A,A)F)!∀!1-!2!+-!-.(14),B#A)F,B~#B)F,!(B~)=!+(B~)-!-(B~)∗!+(B~)∗!+(B)=!(B)Dc)=!(B~)=!+(B~)-!-(B~)∀-!-(B~)∀-!-(B)!(B)D)=(38)(39)B#A)F,!+(B)=supB~&B,B~#A)F!(B~)=!-(B)=-infB~&B,B~#A)F!(B~)=-!=!+-!-(A,A)F)!Jordan,A=A)Dc+A)DHahn,%!=!++!-!.,!~+!~-(,F).A#F,A)n#F!2(A)n