《实分析与复分析》答案

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1、课后答案网www.khdaw.com
chapterone3.ProvethatiffisarealfunctiononameasurablespaceXsuchthatfx:f(x)rgismeasurableforeveryrationalr,thenfismeasurable.[proof]:Foreachrealnumber,thereexistsandescendingsequencefrngofrationalnumberssuchthatlimrn=.Moreover,w

2、ehaven!1[1(;+1)=[rn;+1):n=1Hence,[111f((;+1))=f([rn;+1)):n=1Sincesetsf1([r;+1))aremeasurableforeachn,thesetf1((;+1))isnalsomeasurable.Thenfismeasurable.4.Letfangandfbngbesequencesin[1;+1],andprovethefollowingassertions:(a)limsup(an)=liminfan:n!1n!1

3、(b)limsup(an+bn)limsupan+limsupbn:n!1n!1n!1providednoneofthesumsisoftheform11.(c)Ifanbnforalln,thenliminfanliminfbn:n!1n!1Showbyanexamplethatstrictinequalitycanholdin(b).[proof]:(a)Sincesup(ak)=infak;n=1;2;


:knknTherefore,letn!1,itobtainslimsup(

4、ak)=liminfak:n!1knn!1kn1课后答案网www.khdaw.comBythedenationsoftheupperandthelowerlimits,thatislimsup(an)=liminfan:n!1n!1(b)Sincesup(ak+bk)supak+supbk;n=1;2;


:knknknHencelimsup(ak+bk)lim[supak+supbk]=limsupak+limsupbk:n!1knn!1knknn!1knn!1knB

5、ythedenationsoftheupperandthelowerlimits,thatislimsup(an+bn)limsupan+limsupbn:n!1n!1n!1example:wedenenn+1an=(1);bn=(1);n=1;2;


:Thenwehavean+bn=0;n=1;2;


:Butlimsupan=limsupbn=1:n!1n!1(c)Becauseanbnforalln,thenwehaveinf(ak)infbk;n=1;2;


:knknB

6、ythedenationsofthelowerlimits,itfollowsliminfanliminfbn:n!1n!15.(a)Suposef:X![1;+1]andg:X![1;+1]aremeasurable.Provethatthesetsfx:f(x)

7、anitelimit)ismeasurable.2课后答案网www.khdaw.com[proof]:(a)Sinceforeachrationalr,thesetfx:f(x)

8、t11fx:f(x)=g(x)g=fx:jf(x)g(x)j