實(shí)變函數(shù)與泛函分析基礎(chǔ)第三版(程其襄) 課后答案

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1.?\A∪(B∩C)=(A∪B)∩(A∪C).

XUux∈(A∪(B∪C)).px∈A,9x∈A∪B,x∈A∪C,)x∈(A∪B)∩(A∪C).

px∈B∩C,9!0x∈A∪Bdx∈A∪C,x∈(A∪B)∩(A∪C),*

A∪(B∩C)?(A∪B)∩(A∪C).

V$-[ux∈(A∪B)∩(A∪C).px∈A,m0x∈A∪(B∩C).px/∈A,/x∈A∪Bdx∈A∪C,Ox∈Bdx∈C,%x∈B∩C,!0x∈A∪(B∩C),*(A∪B)∩(A∪C)?A∪(B∩C).%A∪(B∩C)=(A∪B)∩(A∪C).

2.?\

(1)A?B=A?(A∩B)=(A∪B)?B;

(2)A∩(B?C)=(A∩B)?(A∩C);

(3)(A?B)?C=A?(B∪C);

(4)A?(B?C)=(A?B)∪(A∩C);

(5)(A?B)∩(C?D)=(A∩C)?(B∪D);(6)A?(A?B)=A∩B.

XU(1)A?(A∩B)=A∩?s(A∩B)=A∩(?sA∪?sB)=(A∩?sA)∪(A∩?sB)=A?B;

(A∪B)?B=(A∪B)∩?sB=(A∩?sB)∪(B∩?sB)=A?B;

(2)(A∩B)?(A∩C)=(A∩B)∩?s(A∩C)=(A∩B)∩(?sA∪?sC)=(A∩B∩?sA)∪(A∩

B∩?sC)=A∩(B∩?sC)=A∩(B?C);

(3)(A?B)?C=(A∩?sB)∩?sC=A∩?s(B∪C)=A?(B∪C);

(4)A?(B?C)=A?(B∩?sC)=A∩?s(B∩?sC)=A∩(?sB∪C)=(A∩?sB)∪(A∩C)=(A?B)∪(A∩C);

(5)(A?B)∩(C?D)=(A∩?sB)∩(C∩?sD)=(A∩C)∩?s(B∪D)=(A∩C)?(B∪D);(6)A?(A?B)=A∩?s(A∩?sB)=A∩(?sA∪B)=A∩B.

3.?\(A∪B)?C=(A?C)∪(B?C); A?(B∪C)=(A?B)∩(A?C).

XU (A∪B)?C=(A∪B)∩?sC=(A∩?sC)∪(B∩?sC)=(A?C)∪(B?C);

(A?B)∩(A?C)=(A∩?sB)∩(A∩?sC)=A∩?sB∩?sC=A∩?s(B∪C)=A?(B∪C).

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Si))=

i=1

m(T∩Si)=

i=1

mEi.

5.am?E=0,ED

" _61T,T=(E∩T)∪(T∩?E),mm?T≤m?(E∩T)+m?(T∩?E).

E∩T?E,mm?(E∩T)≤m?E=0.T∩?E?T,m?(T∩?E)≤m?T,m

m?(E∩T)+m?(T∩?E)≤m?T.

mm?T=m?(T∩E)+m?(T∩?E),$EiD

6.TBt%(Cantor)61!wN

" Bt%6Pi[0,1]bu?Dj*3z=A[;$u?A[

;!w1,&u?A[;!w2,······,nu?A[;!w

3

2n?1

9

Σ∞2n?1

3n,······.P[0,1]b6!w

n=1

3n=1(!DjD:~).

m[0,1]=m(P∪([0,1]?P))=mP+m([0,1]?P).

mmP=m[0,1]?m([0,1]?P)=1?1=0,8Bt%61!w0.

7.A,B?RpXm?B<+∞.aAiD6Tm?(A∪B)=mA+m?B?m?(A∩B).

" AiD6Fn#Ir<

m?(A∪B)=m?((A∪B)∩A)+m?((A∪B)∩?A)=mA+m?(B?A).

O)S

m?B=m?(B∩A)+m?(B∩?A),

m?B<+∞,mm?(B∩?A)<+∞,im?(B?A)=m?B?m?(A∩B),`Vh

m?(A∪B)=mA+m?B?m?(A∩B).

8.TaED"_?>0,2A6G76F,gF?E?G,$

m(G?E)<?,m(E?F)<?.

"mE<∞e"_

Σ∞ S∞

?>0,

MA[;{Ii},i=1,2,···,g

Σ∞

S∞

i=1

Ii?E,X

Σ∞

i=1

|Ii|<mE+?.PG=

i=1

Ii,GwA6G?E,XmE≤mG≤

i=1

mIi=

i=1

|Ii|<

mE+?,mG?mE<?,$m(G?E)<?.

mE=∞eEDwDj*3z=?D6/E=

S∞

n=1

En(mEn<∞),

2n

S

"R*EnbS>,DA6Gn,gGn?EnXm(Gn?En)<?.PG=

S∞

i=1

Gn,G

wA6G?E,XG?E=

S∞

Gn?

S∞

En?

∞

(Gn?En).

n=1

n=1

[∞

n=1

m(G?E)≤

n=1

m(Gn?En)<?.

EDe?EDm"_?>0A6G,G??E,Xm(G??E)<?.

G??E=G∩E=E∩?(?G)=E??G,

PF=?G,Fi6Xm(E?F)=m(G??E)<?.

9.E?Rq,LMD6{An},{Bn},gAn?E?BnXm(Bn?An)→0(n→∞),

ED

""_i,

T∞

n=1

Bn?Bi,m

T∞

n=1

Bn?E?Bi?E.E?Ai,Bi?E?Bi?Ai,m

"_i,

\∞

m?

n=1

!

Bn?E

? ?

≤m(Bi?E)≤m(Bi?Ai)=m(Bi?Ai).

Pi→∞,m(Bi?Ai)→0,m?

∞

T∞

n=1

Bn E

?

=0.m

T∞

n=1

Bn?EiDBnD

T

n=1

BniDmE=

T∞

n=1

Bn?

T∞

n=1

Bn?E

iD

10.A,B?Rp,TJh

m?(A∪B)+m?(A∩B)≤m?A+m?B.

"am?A=+∞4m?B=+∞,>QJm?A<+∞Xm?B<+∞e\

Gδ}6G1?G2,gG1?A,G2?B,?XmG1=m?A,mG2=m?B.

m?(A∪B)≤m(G1∪G2),m?(A∩B)≤m(G1∩G2).

mUp

m?(A∪B)+m?(A∩B)≤m(G1∪G2)+m(G1∩G2)=mG1+mG2=m?A+m?B.

11.E?Rp.a"_?>0,6F?E,gm?(E?F)<?,TEiD

6

"

? 1 S∞

r<"_0 jn, 6Fn?E,gm(E?Fn)<n.PF=

n=1

Fn,

FiD6XF?E."Wjn,

m?(E?F)≤m?(E?Fn

1

)<n.

+m?(E?F)=0,mE?FiD6

E=F∪(E?F)

iD6

12.TybmD61Gμ5jybm61G5j

" yb6^qwM,μ?M,μ≤M.Bt%6i5jwcN!

6$Bt%6W6v!wNiD$yb^q5jc,Bt

%6W6?ybW6i"μ≥M,mμ=M.

Vb'k?9mAd/k

%& "$

1.Pf(x)E_uB/kr<h#[Ekr,5E[f>r]B\.

5E[f=r]Bvf(x)h)B

'# ?#[Ekr,E[f>r]B#[dkα,:{rn}hαTEk

E[f>a]=

S∞

n=1

E[f>rn],

E[f>rn]BE[f>α]hBn

f(x)hE_

B/k

?#[Ekr,E[f=r]Bf(x)hBG\E=(?∞,∞),zh(?∞,∞)

√

B5#[x∈z,f(x)=√3;x/∈z,f(x)=√2,#[Ekr,E[f=r]=?hB

%E[f> 2]=zhBfhB

2.`f(x),fn(x)(n=1,2,···)hW;[a,b]_d/kkuki

\∞

limE

1

|fn?f|<k

k=1n→∞

hEefn(x)jJf(x)5

'# `AuEfnjJ5#[x∈A,[k,N,en>N

|fn(x)?f(x)|<1,

k

k[|

x limE

∈

n→∞

∈

\∞

|fn

1

—f|<k.

1

x limE

k=1n→∞

|fn?f|<k.

T∞ 1

h i

1 1

#[x∈ limE|fn?f|<k,#[?>0,k0,ek0<?,x∈limE

|fn?f|<k0

k=1n→∞

BN,en>Nx∈E

h

1

|fn?f|<k0

i n→∞

k0

,7|fn(x)?f(x)|<1

<?,n

lim

n→∞

fn(x)=f(x),7x∈A.n

\∞

A=

limE

1

|fn?f|<k.

k=1n→∞

3.`{fn}uE_B/kKPojJ5?&^5!hB

§

'#1E6,lim

∞n

n→∞

fn(x)0lim

n→∞

fn(x)!hE_B/kyZE[lim

n→∞

fn=+∞]

hfjJ+ n5E[lim

n→∞

fn=?∞]hfnjJ?∞n5

E[lim

n→∞

fn>lim

n→∞

n→∞

fn]hfnjJn5fn(x)E_jJn5

u

?

E F[lim

n→∞

f=+ ] E[lim

∞?n

n→∞

fn=?∞]?E[lim

fn>limfn].

%hB5%&^5uE[lim

n→∞

fn=+ ] E[lim

∞∪

n→∞

fn= ] E[lim

?∞∪

n→∞

fn>lim

n→∞

fn]

hB5

4.`Eh[0,1]B5M

(

f(x)=

x, x∈E,

?x, x∈[0,1]?E.

vf(x)[0,1]_h)B|f(x)|h)B

!f(x)B?0∈E,E[f≥0]=EB?0/∈E,E[f>0]=EBnf(x)B

x∈[0,1]|f(x)|=xhI~/kn|f(x)|[0,1]_hB

5.`fn(x)(n=1,2,···)hE_a.e.zB/kK%|fn|a.e.jJz/kf.

#[?>0?kc?B5E0?E,m(E\E0)<?,eE0_#Tn|fn(x)|≤c.

FmE<∞.

S∞

'# pE[|fn|=∞],E[fn/→f]!hL5n=0,1,2,···.ME1=E[fn/→

∪

f](

n=0

E[|fn|=∞]),mE1=0.E?E1_fn(x)!zUjJf(x).ME2=E?E1,

n

[x∈E2,sup|fn(x)|<∞.

[∞

E2= E2[sup|fn|≤k],E2[sup|fn|≤k]?E2[sup|fn|≤k+1].

k=1 n n n

k→∞

n

n

nmE2=limmE2[sup|fn|≤k].k0emE2?mE2[sup|fn|≤k0]<?.M

n

E0=E2[sup|fn|≤k0],c=k0.E0_#[n,|fn(x)|≤c,%

m(E?E0)=m(E?E2)+m(E2?E0)<?.

6.`f(x)h(?∞,∞)_I~/kg(x)u[a,b]_B/kf(g(x))hB/

k

S∞

'# :E1=(?∞,∞),E2=[a,b].f(x)E1_I~-#[dkc,E1[f<c]h

{_5`E1[f>c]= (αn,βn),S(αn,βn)hS,W;(BQhz+αnBQu

n=1

?∞,βnBQu+∞).E2[f(g)>c]=

S∞

E2[αn<g<βn]=

S∞

(E2[g>αn]∩E2[g<βn]),

n=1 n=1

ugE2_BE2[g>αn],E2[g<βn]!B-E[f(g)>c]B

7.`/kKfn(x),(n=1,2,···)=5E_”3_”jJf(x),P{fn}a.e.

jJf.

'#ufn(x)E_”3_”jJf(x),n#[δ>0,B5Eδ?E,em(E?Eδ)<δ%fnEδ_jJf(x).`E0hEfnjJYq#[δ,E0?E?Eδ(uEδ_fnjJ),nmE0≤m(E?E0)<δ,Mδ→0,mE0=0.nfn(x)E_a.e.jJf(x)(=r<).

8.iN>EREH

'# N>EREuf(x)hE_/k#[δ>0,5Eδ?Ee

f(x)Eδ_hI~/kUm(E?Eδ)<δ,f(x)hE_a.e.zB/k

#[1/n,5En?E,ef(x)En_I~U

m(E?En

S∞

)<1.

n

S∞ 1

ME0=E?

n=1

En,#[n,mE0=m(E?

n=1

En)≤m(E?En)<n.Mn→∞,

S

S∞

mE0=0.E=(E?E0)∪E0=(

En)∪E0=

∞

En.#[dka,E[f>a]=E0[f>

n=1

S∞

n=0

? ?

∪

(

n=1

En[f>a]),fEn_I~BEn[f>a]B%m(E0[f>a])≤mE0=0,

nE0[f>a]B%E[f>a]hBfhBufEn_z-

S∞

n=1

En_znf(x)a.e.z

9.`/kK{fn}E_"jJf,Ufn(x)≤g(x)a.e.E,n=1,2,···.i

f(x)≤g(x)E_82H

'# ufn(x)?f(x),{fni}?{fn},efni(x)E_a.e.jJf(x).`E0h

S

∞

fni(x)jJf(x)5En=E[fn>g].mE0=0,mEn=0.m(

Σ∞

En)≤

mEn=0.

n=0

S∞

n=0

S∞

?

E

n=0

En_fni(x)≤g(x),fni(x)jJf(x),nf(x)=limfni(x)≤g(x)E?

En

n=0

_H7f(x)≤g(x)E_82H

10.`E_fn(x)?f(x),Ufn(x)≤fn+1(x)82Hn=1,2,···,82

fn(x)jJf(x).

'# ufn(x)?f(x),{fni}?{fn},efni(x)E_a.e.jJf(x).`E0

hfni(x)jJf(x)5En=E[fn<fn+1],mE0=0,mEn=0.

[∞ Σ∞

m( En)≤

mEn=0

n=0

S∞

n=0

?

E

n=0

En_fni(x)jJf(x),Ufn(x)hfn(x)jJf(x).(}K

S∞

KjJ}KajJs4z).7X+L5

?hfn(x)a.e.jJf(x).

n=0

Entfn(x)jJf(x),

11.`E_fn(x)?f(x),%fn(x)=gn(x)a.e.Hn=1,2,···,gn(x)?f(x).

S

Σ

'#`En=E[fn

∞

gn]m(

En)≤

∞

mEn=0.#[σ>0,E[|f?gn|≥σ]?

S

∞

( En)∪E[|f?fn|≥σ].n

n=1

n=1

n=1

[∞

mE[|f?gn|≥σ]≤m(

En)+mE[|f?fn|≥σ]=mE[|f?fn|≥σ].

n=1

ufn(x)?f(x),n0≤limmE[|f?gn|≥σ]≤limmE[|f?fn|]≥σ=07gn(x)?f(x).

j→∞

12.`mE<+∞,PE_fn(x)?f(x)r<h#{fn}[1/kK{fnk},{fnk}/kK{fnkj},elimfnkj(x)=f(x),a.e.E.

'# |DlEH7Bw(|?{fn(x)}E_"jJ

f(x).η0>0,ekK{mE[|fn?f|≥η0]}jJL-k?0>0,6

/kK{fnk},e

mE[|fnk?f|≥η0]>?0>0. (1)

/kK{fnk}82jJf(x)/kKgd_?/kK{fnkj}

E_a.e.jJf,mE<+∞,C*EE_fnkj?f(x),?(1)fO$

13.`mE<∞,82zB/kKfn(x)0gn(x),n=1,2,···,("jJf(x)0g(x),P

fn(x)gn(x)?f(x)g(x);

(2)fn(x)+gn(x)?f(x)+g(x);

(3)min{fn(x),gn(x)}?min{f(x),g(x)};max{fn(x),gn(x)}?max{f(x),g(x)}.

'# (1)f(x)a.e.znmE[|f|=∞]=0.

T∞

n=0

E[|f|≥n]=E[|f|=∞],UE[|f|≥n]?E[|f|≥n+1]0E[|f|≥1]?E,mE[|

f|≥1]≤mE<∞,n

sE

mE[f= ]=lim

| | ∞

n→∞

mE[|f|≥n]=0.

lim

n→∞

mE[|g|≥n]=0.

#[?>0,σ>0,k,mE[|f|≥k]<?0mE[|g|≥k]<?sH

5 5

Mσ0=min

σ

2(k+1)

,fn?f,gn?g,N,en>NmE[|gn?g|≥σ0]<

,1

?,mE[|fn?f|≥σ0]<?sH

5 5

E[|gn|≥k+1]?E[|g|≥k]∪E[|gn?g|≥1]?E[|g|≥k]∪E[|gn?g|≥σ0].

mE[|g|≥k+1]≤mE[|g|≥k]+mE[|g

? ? 2?

—g|≥σ]< + = .

n

h σi

n 0 5 5 5

σ

E|gnfn?gnf|≥2

?E[|gn|≥k+1]∪E

|fn?f|≥2(k+1)?E[|gn|≥k+1]∪E[|fn?f|≥σ0].

n

h σi

2? ? 3?

mE|gnfn?gnf|≥2

%

≤mE[|gn|≥k+1]+mE[|fn?f|≥σ0]<5+5=5.

h σi σ

E|fgn?fg|≥2?E[|f|≥k+1]∪E|gn?g|≥2(k+1)

?E[|f|≥k]∪E[|gn?g|≥σ0].

n h

mE|fg

σi

—fg|≥

≤mE[|f|≥k]+mE[|g

? ? 2?

—g|≥σ]< + = .

n 2 n 0

h σi h

5 5 5

σi

E[|gnfn?gf|≥σ]?E

n

h

|gnfn?gnf|≥2

σi

∪E|fgn?fg|≥2,

h σi 3? 2?

mE[|gnfn?gf|≥σ]≤mE

|gnfn?gnf|≥2

+mE

|fgn?fg|≥2

< + =?.

5 5

7#[?>0,σ>0,N,en>NmE[|gnfn?gf|≥σ]<?,ngnfn?gf.

(2)

E[|(fn+gn)?(f+g)|≥σ]?E

n

h σi h σi

|fn?f|≥2∪E|gn?g|≥2

h σi h σi

mE[|(fn+gn)?(f+g)|≥σ]≤mE

|fn?f|≥2

h

+mE

σi

|gn?g|≥2,

h σi

lim

n→∞

mE[|(fn+gn)?(f+g)|≥σ]≤limmE

|fn?f|≥2

+limmE

n→∞

|gn?g|≥2.

n→∞

7fn+gn?f+g.

(3)x?fn?f,|fn|?|f|.gd_

E[|fn?f|≥σ]?E[||fn|?|f||≥σ].

nlim

n→∞

mE[||fn|?|f||≥σ]≤lim

mE[|fn?f|≥σ]=0,7|fn|?|f|.

n→∞

?fn?f,#[a/=0,afn?af.gd_

σ

E[|afn?af|≥σ]=E

n

|fn?f|≥|a|,

n→∞

σ

lim

n→∞

mE[|afn?af|≥σ]=limmE

|fn?f|≥

|a|

=0.

(2),

min{fn

(x),gn

(x)=fn(x)+gn(x)?|fn(x)?gn(x)|.

}

2

%

(2),

n

fn(x)+gn(x)?f+g,fn(x)?gn(x)?f?g.

|fn(x)?gn(x)|?|f(x)?g(x)|,

fn(x)+gn(x)?|fn(x)?gn(x)|?f(x)+g(x)?|f(x)?g(x)|.

2

2

fn(x)+gn(x)?|fn(x)?gn(x)|?f(x)+g(x)?|f(x)?g(x)|.

7

s

min{fn(x),gn(x)}?min{f(x),g(x)}.

max{fn

(x),gn

(x)=fn(x)+gn(x)+|fn(x)?gn(x)|,

}

2

max{fn(x),gn(x)}?max{f(x),g(x)}.

s~.0/IU0<

FLM HGJ

1.(-Lebesgue$%{D2*(-Darboux%\Oh3_

I(Darboux%\(}f(x)E{+PV<(Ex>V2B

··· →

D:E1,E2, ,En,maxmEi 0

1≤i≤n

Oh"%_

S(D,f)→

,ˉ ,

→

f(x)dx,S(D,f)

E

?

Ef(x)dx.

^yT[0,1]uL{"?WZ<

1, x[0,1]B+\,

f(x)=

0, x[0,1]B\.

.i?1i n .n?1 .

(j798n,E[0,1]uL2BD={En},oBEn= , ,i=1,2,···,n?1,E= ,

n i

3maxmEn=1→0(n→∞).

i nn

n n1.

1≤i≤n i n

Xn

S(D,f)=

Xn

i

supmEn=

1·1

=1.

i=1

x∈En

i

n

i=1

f"/k&f(x)2[0,1]{Y6VD9

,ˉ

&(Darboux%\_

[0,1]

,

f(x)dx=

[0,1]

f(x)dx=0.

3n

2.}2CantorEP0{%%<f(x)=0,*2P0.EB18uL{%%

n(n=1,2,···),:f(x)VD2?tD2=

3n

NK f(x)14V<&*D2w%>:lD2+GV}EnP0.E

3n

B 1

8uL<?3mEn

=2n?1,&

,

[0,1]

f(x)dx=

X∞,

n=1En

f(x)dx=

X∞

n=1

nmEn=

X∞

·

n

n=1

2n?1

=3

3n .

#f(x)VDrD2=3.

3.}f(x)2E{VDen=E[|f|≥n],3

·

limnmen=0.

n

NK *-f(x)2E{VD9E{a.e.+V<#mE[|f|=∞]=0.f"

n=1

/k*en?en+1,me1≤mE<∞#FT∞en=E[|f|=∞],

n

limmen=mE[|f|=∞]=0.

,

*-|f(x)|VD*D2R(`(-x$?>0,2δ>0,e?Erme<δ

e

|f(x)|dx<?.

(δ>0,2N,n>Nmen<δ,9

,

·

Glimnmen=0.

n

n·men≤

f(x)dx<?.

| |

en

4.}mE<∞,f(x)E{V<En=E[n?1≤f<n],3f(x)2E{VDM

?∞

.∞|n|mEn<∞.

NK :zf(x)2E{VD3|f(x)|2E{VD

n≥12En{n?1≤|f(x)|=f(x)<n.

n≤02En{|n|≤|f|≤|n?1|=1?n,&

,

>

E

∞ |f(x)|dx=

X∞,

n=1En

|f|dx+

X?∞,

n=0En

|f|dx≥

X∞

n=1

(n?1)mEn+

X?∞

n=0

|n|mEn

=X∞

n=1

|n|mEn+

X?∞

n=0

|n|mEn?

X∞

n=1

X∞

mEn=

?∞

|n|mEn?

X∞

n=1

mEn,

&EnbbNECE

X∞

n=1

mEn=m(

X∞

n=1

En)≤mE<∞,

?∞

&.∞|n|mEn<∞.

| |

1:2z.∞nmEn,3

?∞

,

E

|f(x)|dx=

X∞,

n=1En

|f|dx+

X?∞,

n=0En

|f|dx≤

X∞

n=1

nmEn+

X?∞

n=0

|n?1|mEn

=X∞

n=1

|n|mEn+

X?∞

n=0

|n|mEn+

X?∞

n=0

X∞

mEn≤

?∞

|n|mEn+mE<∞.

#|f