Các dạng hội tụ của dãy hàm đo được
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Transcript of Các dạng hội tụ của dãy hàm đo được
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Hunh Vit Khnh SP. Ton 01-K.30 - 1 -
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LI NI UCc dng hi t ca dy hm o c l mt phn
nh trong lnh vc o v tch phn Lebesgue. y l
mt trong cc mng gii tch c ng dng nhiu trongthc t, v c bit l nn tng cho gii tch hin i. Do
, vic nghin cu v n l rt cn thit.
V thi gian hon thnh lun vn ny tng i
ngn nn khng th nghin cu su hn, v chc cn
nhiu sai st. Rt mong nhn c s gp ca qu thy
c v qu bn c.
Em xin chn thnh cm n B mn Ton to
iu kin cho em nghin cu. Xin cm n c Trn Th
Thanh Thy nhit tnh hng dn v gip em sa cha
kp thi cc sai st trong lun vn ny.
Sinh vin thc hin
Hunh Vit Khnh
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Hunh Vit Khnh SP. Ton 01-K.30 - 3 -
NHN XT CA GIO VIN HNG DN
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Cn Th, ngy thngnm 2008
Trn Th Thanh Thy
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Hunh Vit Khnh SP. Ton 01-K.30 - 4 -
NHN XT CA GIO VIN PHN BIN
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Cn Th, ngy.. thng.. nm 2008
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Hunh Vit Khnh SP. Ton 01-K.30 - 5 -
MC LC
MU................................................................................................................. 71. L do chn ti.............................................................................................. 72. Gii hn ca ti............................................................................................ 7
3. Mc tiu ti.................................................................................................. 7
NI DUNG ............................................................................................................. 9
Chng 1: KIN THC CHUN B........................................................................ 91.1 O.......................................................................................................... 9
1.1.1 i s tp hp.......................................................................................... 91.1.2. - i s ................................................................................................. 91.1.3. - i s Borel...................................................................................... 101.1.4. o trn mt i s tp hp ............................................................... 111.1.5 Mrng o ....................................................................................... 13
1.1.6 o trn r ......................................................................................... 151.2- HM SO C .................................................................................. 171.2.1 nh ngha ............................................................................................. 171.2.2 Mt s tnh cht ca hm so c ..................................................... 181.2.3 Cc php ton trn cc hm so c.................................................. 20
1.3- TCH PHN LEBESGUE .......................................................................... 231.3.1. Tch phn ca hm n gin khng m ................................................. 231.3.2 Tch phn ca hm o c khng m................................................... 241.3.3 Tch phn ca hm o c bt k ......................................................... 261.3.4 Tnh cht................................................................................................ 26
1.3.5 Gii hn qua du tch phn..................................................................... 27Chng 2: SHI TCA DY HMOC ............................................. 30
2.1 CC DNG HI T CA DY HM O C.................................... 302.1.1Hi t hu khp ni (converges almost everywhere) .............................. 302.1.2Hi tu (converges uniformly) ........................................................... 312.1.3Hi tu hu khp ni (converges uniformly almost everywhere)........ 322.1.4Hi t theo o (converges in measure) .............................................. 322.1.5Hi t trung bnh (converges in the mean) ............................................. 342.1.6Hi t hu nhu (converges almost uniformly) .................................. 35
2.2 CC DNG DY C BN ........................................................................ 36
2.2.1Dy cbn hu khp ni (Cauchy almost everywhere, hoc fundamentalalmost everywhere)......................................................................................... 362.2.2 Dy cbn u ( uniformly Cauchy)..................................................... 372.2.3 Dy cbn hu nhu (almost uniformly Cauchy)............................. 372.2.4 Dy hm cbn trung bnh (Cauchy in the mean hoc mean fundamental)....................................................................................................................... 372.2.5Dy cbn trong o (Cauchy in measure, hoc fundamental inmeasure)......................................................................................................... 37
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2.3 SLIN H GIA CC DNG HI T CA DY HM O C...382.3.1 Lin h gia hi t trung bnh v hi t theo o ................................ 382.3.2 Lin h gia hi t trung bnh v hi t hu khp ni ........................... 392.3.3 Lin h gia hi t theo o v hi t hu khp ni ............................ 402.3.4 Lin h gia hi t trung bnh v hi tu........................................... 43
2.3.5 Lin h gia hi t hu nhu v hi t hu khp ni ......................... 432.3.6 Lin h gia hi t theo o v hi t hu nhu ............................. 452.3.8 Lin h gia hi t hu khp ni v hi tu....................................... 482.3.9 Lin h gia hi t trung bnh v cbn trung bnh............................... 492.3.10 Lin h gia cbn trung bnh v cbn theo o........................... 502.3.11 Lin h gia cbn trung bnh v hi t hu nhu.......................... 502.3.12 Lin h gia cbn hu nhu v hi t hu nhu....................... 502.3.13 Lin h gia cbn theo o v cbn hu nhu........................ 522.3.14 Lin h gia cbn theo o v hi t theo o ............................ 532.3.15 Lc th hin mi lin h gia cc dng hi t................................ 54
Chng 4: BI TP............................................................................................... 56KT LUN........................................................................................................... 72
TI LIU THAM KHO........................................................................................ 73
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MU
1. L do chn ti
o v tch phn Lebesgue l nn tng ca gii tch hin i. Vic nghin
cu n l cn thit, gip cho em nm vng hn kin thc v phn ny. Ngoi ra, em
cn c iu kin nghin cu su hn cc mng gii tch c lin quan. y l l do
chnh em chn ti ny.
2. Gii hn ca ti
o v tch phn Lebesgue l mng gii tch hin i kh rng. Trong
khung kh mt lun vn tt nghip, ti khng th khai thc mi vn . Do vy,
lun vn tp trung khai thc v mt s dng hi t ca dy hm o c. Bn cnh
, cn xt v mi lin h gia cc dng hi t ny.
3. Mc tiu ti
Trong phm vi gii hn ca ti, mc tiu hng ti ca lun vn l nghin
cu mt s dng hi t ca dy hm o c. C th hn, bn cnh cc dng hi t
quen thuc nhhi t theo o, hi t hu khp ni, ti cn nghin cu mt s
dng hi t khc nh hi t hu nhu, hi tu hu khp ni, hi t trung
bnh,
Tuy nhin, hiu su hn v cc dng hi t, ti cn tp trung nghin
cu v mi lin h gia cc dng hi t ny. V d, nh ta bit, trong khng gian
o hu hn v o c xt l o th mi dy hm o c hi t hu
khp ni th hi t theo o. Vn t ra l i vi cc dng hi t khc th c
mi lin h vi nhau nh th no? V cc mi lin h ny c thay i hay khng khi
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Hunh Vit Khnh SP. Ton 01-K.30 - 8 -
ta xt chng trong khng gian o hu hn? ti s tp trung lm r cc vn
ny.
thun tin trong qu trnh nghin cu, lun vn cn cp n mt s
khi nim mi nh dy cbn theo o, dy cbn trung bnh,V khng ngoil, lun vn cng cp n mi lin h gia cc khi nim ny.
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Hunh Vit Khnh SP. Ton 01-K.30 - 9 -
NI DUNG
Chng 1: KIN THC CHUN B.
1.1 O
1.1.1 i s tp hp
nh ngha
Mt i s (hay trng) l mt lp nhng tp cha X, v kn i vi miphp ton hu hn v tp hp (php hp, php giao hu hn cc tp hp, php hiu
v hiu i xng hai tp hp).
nh l 1
Mt lp tp hp l mt i skhi v ch khi C tha mn cc iu kin sau:
a. C ;
b. A C CA C ;
c. BA, C BA C .
1.1.2. - i s
nh ngha
Mt - i s (hay - trng) l mt lp tp hp cha ,A v kn i vi mi
php ton m c hay hu hn v tp hp.
nh l 2
Mt lp tp hp F l mt -i skhi v ch khi F tha mn cc iu kin sau:
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Hunh Vit Khnh SP. Ton 01-K.30 - 10 -
a. F ;
b. A F CA F ;
c. nA F 1
nn
A
=
U F .
Nhn xt
Mt - i shin nhin l mt i s.
nh l 3
Cho M l mt h khng rng cc tp con ca X.
a. Lun tn ti duy nht mt i s ( )C M bao hm M v cha trong tt c
cc i skhc bao hm M i s ( )C M gi l i ssinh bi M .
b. Lun tn ti duy nht mt - i s ( )F M bao hm M v cha trong tt
c cc - i s khc bao hm M - i s ( )F M c gi l - i s
sinh bi M .
1.1.3. - i s Borel
nh ngha
Cho khng gian tp ,(X ) . - i s sinh bi h tt c cc tp m trong X
c gi l - i sborel.
K hiu: ( )XB .
Nhn xt
Cc tp m, tp ng l cc tp Borel.
Nu , 1,2,...nA n = l cc tp Borel th1
nn
A
=U v
1n
n
A
=I , theo th t l
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Hunh Vit Khnh SP. Ton 01-K.30 - 11 -
cc tp kiu F ,G cng l nhng tp Borel.
- i sBorel trong mt khng gian tp Xcng l - i snh
nht bao hm lp cc tp ng.
1.1.4. o trn mt i s tp hp
nh ngha
Cho C l mt i strn X.
Hm tp hp : C R l mt o trn C nu:
a. ( ) 0A , A C .
b. ( ) 0; =
c. =
=U
1nnA ( )
1n
n
A
= , vi ( ) , , .n m nA A m n A n = C
o c gi l hu hn nu ( ) +
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Cc tnh cht ca o
nh l 4
Cho l o trn i sC .a. BA, C , ( ) ( )B A B A ;
b. BA, C , ( ) +
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nh l 5
Cho l o trn i sC .
a. Nu ( ) ,0=iA 1 ii A
= U C th .01 =
=Ui iA
b. NuAC , ( ) 0=B th ( ) ( ) ( )\BA B A A = = .
nh l 6
Cho l o trn i sC .
a. Nu nA C ( ),n ....,21 AA1
n
n
A
=
U
C th ( )1
lim .n nnn
A A
=
=
U
b. Nu nA C ( ),n ....,21 AA ( ) ,1 +
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Hunh Vit Khnh SP. Ton 01-K.30 - 14 -
a. ( )* 0, ;A A X
b. ( )* 0; =
c.1
n
n
A A
=
U ( ) ( )* *
1
n
n
A A
=
(tnh cht - bn cng tnh).
nh l 8 (nh l Carathodory)
Cho * l mt o ngoi trn ,X L l lp cc tp con A caXsao cho:
( ) ( ) ( )* * * \ ,E E A E A E X = + ( )1
Khi :
a. L l mt - i s.b. * = | L l mt o trn L .
o ny c gi l o cm sinh bi o ngoi * .
Tp A tha ( )1 c gi l * -o c.
nh l 9
Cho m l mt o trn mt i sC
nhng tp con ca X.Nu vi mi A X t:
( )* infA = ( )1 1
, ,i i ii i
m P P A P
= =
U C ( )2
Khi :
a. * l o ngoi;
b. * | L m= ;
c. C ( )F C .L
nh l 10
Nu l cm sinh bi o ngoi * th:
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Hunh Vit Khnh SP. Ton 01-K.30 - 15 -
a. H cc tp c o bng 0 trng vi h tp c o ngoi *
bng 0.
b. l o .
nh l 11 (mrng o)
Cho m l o trn mt i s L Khi , tn ti mt o trn - i s
L ( )F C C , sao cho:
a. ( ) ( );A m A =
b. l hu hn (- hu hn) nu m l hu hn (- hu hn);
c. l o ;d. A L khi v ch khiA biu din c di dng:
\A B N = hoc A B N =
Trong B ( )F C , N E ( )F C , ( ) ( )* 0E E = = v * l o ngoi xc
nh t m bi cng thc ( )2 .
Nhn xt
L sai khc ( )F C mt b phn cc tp c o khng, tc l - i s L cc tpo c c th thu c t ( )F C bng cch thm hay bt mt b phn ca mt tp
c o khng.
1.1.6 o trn r
Ta gi gian trn ng thng l mt tp hp c mt trong cc dng
sau:[ ] ( )( ] [ ) ( ) ( ) ( ) ( ] [ ), , , , , , , , , , , , , , , ,a b a b a b a b a a a a + + + .
Xy dng i s
Gi C l lp tt c cc tp con ca c th biu din thnh hp ca mt s hu
hn cc gian i mt ri nhau, tc l:
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Hunh Vit Khnh SP. Ton 01-K.30 - 16 -
( )1
: , ,n
i i ii
P P i j n=
= =
UC r= r r r .
Khi , C l mt i s.
Nu P C v ( )1
, ,
n
i i ji
P i j=
= = Ur r r t ( ) 1
n
ii
m P== r .
Khi , m l o trn C v m l o - hu hn.
Mrng o
Vi ,A R o ngoi c xc nh bi:
( ) ( )*
1 1
inf , ,i i ii i
A m P P A P
= =
=
U C
iu ny c th thay bng:
( )*1 1
inf , ,k k kk i
A A
= =
=
Ur r r
Gi L l tp tt c cc tp con A ca sao cho:
( ) ( ) ( )* * * ,E E A E A E = + \ .
o mrng trn - i s L c gi l o Lebesgue.
Cc tp A L c gi l nhng tp o c theo ngha Lebesgue (hayA o
c ( )L .
Nhn xt
o Lebesgue trn l - hu hn v [ ]1
,n
n n
=
U= v hin nhin
. l o . Mi tp Borel trn u o c Lebesgue.
Tp o c Lebesgue chnh l tp Borel thm hay bt mt tp c .
. o khng.
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Hunh Vit Khnh SP. Ton 01-K.30 - 17 -
1.2- HM SO C
1.2.1 nh ngha
Cho tp ,X F l mt i snhng tp con ca X, v AF .
Khng gian ( ),X F dc gi l khng gian o c.
Mt hm s :f A c gi l o c trn tp A i vi i s F nu:
( ){ }, : .a x A f x a < F
Hay vit gn l:
{ },A
a f a < FR .
Nu trn F c o th f c gi l o c i vi o hay o
c.
Nu ,k=F L v kX = th ta ni f o c theo ngha Lebesgue hay ( )L - o
c.
Nu k=F B , v kX = th ta ni f o c theo ngha Borel hay f l hm s
Borel.
Nhn xt.
Hm s f o c trn A ( )+ ,1 af F , a .
nh l 1
Cho ( ),X F l khng gian o c v hm :f X . Khi , cc iu kin sau
l tng ng.
( )i Hm f o c trn A
( )ii { },A
a f a < F
( )iii { }, Aa f a F
( )iv { },A
a f a > F
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Hunh Vit Khnh SP. Ton 01-K.30 - 18 -
( )v { },A
a f a F
Chng minh
( ) ( )i ii : Do nh ngha.
( ) ( )ii iii : ( ) ( ) 1, ,a f x a f x a nn
< +
{ }1
1.
An A
f a f an
=
= < +
I F
( ) ( )iii iv : ,a t:
{ }A
M f a=
{ }AN f a= <
Ta c:
M N A = , v M N =
N A M = \
N F .
( ) ( )iv v : Ta c:
,a ( ) ( )1
:f x a n f x a
n
>
{ }1
1.
An A
f a f an
=
=
I F
( ) ( )v i : ,a t:
{ }A
D f a=
{ }A
E f a= <
E A D= \ A D= F
Vy fo c.
1.2.2 Mt s tnh cht ca hm so c
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Hunh Vit Khnh SP. Ton 01-K.30 - 19 -
( )i Gi s f o c trn A . Nu B A , B F th f cng o c trnB .
Tht vy, v B A , v B F nn: ,a
{ } { } .B A
f a B f a< = < F
Vy, f o c trn .B
( )ii Nu f o c trn A th { }, .A
a f a = F
Tht vy,
,a R { } { } { }A A A
f a f a f a= =
Do : { }A
f a= F .
( )iii Nu ( ) ,f x c x A= th fo c trn .A
Tht vy, ,a ta thy:
{ }, ;
, .Ac a
f aA c a
< =
F
Nu 0k> th { } .A
akf a f
k < = <
F
Do , vi 0k ta c kf o c trn A .
Nu 0k= th ( )( ) 0, .kf x x A=
Do , theo ( )iii , ta c kf o c trn A .
Nh vy, hm kfo c trn A .
( )v Nu f o c trn { }n nA (hu hn hoc m c) th f o c trn
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Hunh Vit Khnh SP. Ton 01-K.30 - 20 -
j nn
AU .
Tht vy, ,a ta c:
{ } { }n nn A Anf a f a
< = < U UF
Vy, f o c trn .nn
AU
( )vi Nu f xc nh trn A , ( ) 0A = v th f o c trn A .
Tht vy, ,a ta c:
{ }A
f a A<
Do ( ) 0,A = v nn { }Af a< F Vy, f o c trn A .
1.2.3 Cc php ton trn cc hm so c
Cho ( ),X F l khng gian o c, A F .
( )i Nu f o c trn A th vi 0 > , hm f
o c trn A .
Tht vy, vi 0 > , ta c:
{ } { } { } , a 0;
, 0.AA A
f af a f a a
< = > < >
{ }A
f a < F
Vy,
f o c trn .A
Tuy nhin, mnh o ca mnh trn ni chung khng ng. Ngha .l, c th xy ra trng hp f
nhng f khng o c.
V d: Xt hm s:
( )1, ;
1, .
x Af x
x A
=
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Hunh Vit Khnh SP. Ton 01-K.30 - 21 -
Trong ,A , A l mt tp khng o c Lebesgue.
Ta c 11
, .2
f A + =
Do , fkhng o c trn .
Nhng, ( ) 1,f x x= R nn f o c trn .
( )ii Nu f v g o c, hu hn trn A th f g+ o c trn A .
Gi { }n nr l dy cc s hu t.
,a f g a+ < f a g <
: .nn f r a g < <
{ } { } { }( )1 .n nA A Anf g a f r g a r + < < < U F
f g + o c trn .A
Tuy nhin, mnh o ca mnh trn ni chung khng ng. Ngha
l, nu ta c f g+ o c th cha suy ra c f v g o c.
V d: Xt cc hm s
( )
=Ax
Axxf
,0
,1v ( )
1, ;
0, .
x Ag x
x A
=
Vi ,A A l tp khng o c Lebesgue.
Ta c:
( )1 ,0f A =
v ( ) Ag =+ ,01 .
nn gf, l nhng hm s khng o c trn .
Nhng, ( )( ) 0,f g x x+ = R nn gf + o c trn .
( )iii Nu f v g o c v hu hn trn A th f g cng o c trn A .
Tht vy, v g o c nn g o c. Do , ( )f g f g = + o
c trn A .
( )iv Nu f v g o c, hu hn trn A th .f g o c trn A .
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Hunh Vit Khnh SP. Ton 01-K.30 - 22 -
Tht vy, ( ) ( )2 21
.4
f g f g f g = + nn .f g o c trn .A
Tuy nhin, mnh o ca mnh ( )iv khng ng.
V d: Xt cc hm s
( )1, ;
0, .
x Af x
x A
=
v ( )0, ;
1, .
x Ag x
x A
=
Vi ,A A l tp khng o c Lebesgue.
R rng, ,f g khng o c trn .
Nhng, ( )( ). 0,f g x x= nn gf. l hm o c trn .
Nhn xt:
Hm f o cA hmf+ v f o c trn .A
Trong :
{ }max ,0 ;f f+ = { }max ,0f f =
( )v Nu f v g o c, hu hn trn A th { } { }max , , min ,f g f g o c
trn .A
Tht vy, ta c:
{ } ( )1max , 2f g f g f g = + + ;
{ } ( )1
min ,2
f g f g f g = + l nhng hm o c trn .A
Do { }min ,f g , { }max ,f g o c trn A .
( )vi Nu f v g o c v hu hn trn A , ( ) 0, ,g x x A thf
go
. c trn .A
Tht vy, do ( ) 0,g x x A nn:
,a 22
, 0;1
1, 0
AA
a
ag ag
a
< = > >
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Hunh Vit Khnh SP. Ton 01-K.30 - 23 -
2
1.
A
ag
<
F
Nh vy,2
1
go c trn .A
Do2
1.
ff g
g g= nn suy ra
g
fo c trn .A
( )vii Nu dy ( ){ }n nf x N l mt dy nhng hm so c v hu hn trn
A th cc hm s ( ){ }sup n nn
f xN
, ( ){ }inf n nn f x N , ( ){ }lim n nf x N ,
( ){ }lim n nf x N l nhng hm o c, v nu tn ti lim nx f f = , th f
cng o c trn A .Tht vy,
,a R ( ){ }{ } { }1
sup n n An A n
f x a f a
= I F
,a R ( ){ }{ } { }1
inf n n An A nf x a f a
< = < U F
Do , ( ){ }sup n nn
f xN
, ( ){ }inf n nn f x N l nhng hm o c trn .A
V 1lim inf sup ;n mn m nf f = 1lim sup inf n mm nnf f=
Nn suy ra nn ff lim,lim cng l nhng hm o c trn .A
Do , nu ffnn
=
lim th lim nf f=
Vy, f o c trn A .
1.3- TCH PHN LEBESGUE
1.3.1. Tch phn ca hm n gin khng m
nh ngha
Xt mt khng gian c o ( ), ,X F , AF .
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Hunh Vit Khnh SP. Ton 01-K.30 - 24 -
Hm s f xc nh trn A c gi l hm n gin nu fo c v nhn
mt s hu hn nhng gi tr hu hn.
Nh vy, nu f l hm n gin khng m xc nh trn tp .A F Khi , f
c dng:
( )=
=n
iA xaf i
1
( )*
Trong , iA o c, ri nhau v Un
iiAA
1=
= .
Ngi ta gi ( )=
n
iii Aa
1
l tch phn ca hm n gin fi vi o trn .A
K hiu: Afd
.Tch phn ca hm n gin khng m f c xc nh bi ( )* l duy nht vi
mi cch biu din ca hm f .
1.3.2 Tch phn ca hm o c khng m
Trc khi trnh by nh ngha tch phn hm o c khng m, lun vn
cp li nh l v cu trc ca hm o c:
nh l
Mi mt hm so c trn A u l gii hn ca mt dy { }n nf nhng hm
n gin trn A : lim , .nn
f f x A
=
Hn na, nu 0,f th tn ti { }n nf sao cho:
nf n gin, 0nf , 1n nf f+ , v lim , .nn f f a =
Chng minh
Ta chng minh cho trng hp 0f trn .A
Vi mi s t nhin n , ta t:
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Hunh Vit Khnh SP. Ton 01-K.30 - 25 -
( ){ }nxfAxCn = :0
( ) ( )1
: , 1, 2,..., 22 2
i nn n n
i iC x A f x i
= < =
t:
( )
0,
1,
2
n
n inn
n x Cf x i
x C
=
Khi , nf l hm n gin trn A , 0nf , v 1n nf f+
Ta chng minh lim nn
f f
=
+ Nu ( )
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Hunh Vit Khnh SP. Ton 01-K.30 - 26 -
Gi s f l o c khng m, xc nh trn tp A . Khi , tn ti nf
dy hm n gin, khng m, n iu tng, v lim .nx
f f
=
Tch phn ca hm f trn A i vi o c nh ngha l:
fd = lim nnA
f d .
1.3.3 Tch phn ca hm o c bt k
Nu f l hm o c bt k, ta phn tch: + = fff .
NuA
f d+ hocA
f d hu hn th hiu sA
f d+ A
f d c ngha v n
c gi l tch phn ca hm f trn A i vi o .
Hm f c gi l kh tch trn A nu A
fd hu hn.
1.3.4 Tnh cht
( )i Cc tnh cht n gin:
( ).A cd c A = ( ) ( ).B
A
x d A B =
( ) ( )1 1
.i
n n
i B i ii iA
x d A B = =
=
( )ii Nu f o c trn A v ( ) 0A = th 0.A
fd =
( )iii Nu f o c, gii ni trn A v ( )A < th fkh tch trn A .( )iv Tnh cht cng tnh:
Nu A B = thA B A B
fd fd fd
= + , nu mt trong hai v ca
ng thc c ngha.
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( )v Tnh bo ton th t:
Nu gf = h.k.n trn A th .A A
fd gd =
( gf = h.k.n trn A nu ( ) 0: = BAB v ( ) ( ) B\, Axxgxf = ).
Nu f g trn A thA A
fd gd
( )vi Tnh cht tuyn tnh:
,A A
cfd c gd c = .
( ) .A A A
f g d fd gd + = + (nu v phi c ngha).
1.3.5 Gii hn qua du tch phn
nh l hi tn iu
Cho dy hm o c { }nf .
Nu 0 nf f trn A th lim nn
A A
f d fd
= .
Chng minh Nu { }nf l dy cc hm n gin th hin nhin theo nh ngha tch phn ta
c lim .nn
A A
f d fd
=
Xt trng hp { }nf bt k.
Gi ijh l cc hm n gin khng m sao cho :
11131211 ... fhhhh n
22232221 ... fhhhh n
.
t { }1 2max , ,...,n n n nnh h h h=
Ta c: nh l dy hm n gin, khng m, n iu tng v nn fh
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Hunh Vit Khnh SP. Ton 01-K.30 - 28 -
Do , ( )*n nA A A
h f f
Mc khc, nk th nnkn fhh
Cho n ta c : fhf nn
k
lim
Cho k , ta c fhnn
=
lim
Kt hp vi (*), v cho n ta suy ra : lim nn
A A
f d fd
= .
B Fatou
Nu 0nf trn A th lim lim .n nA
f d f d
Chng minh
t { },...,inf 1+= nnn ffg
Ta c nn fg lim0
Do , =A
n
A
nn
fg limlim
Nhng v nn fg , n nn A
n
A
n fg
Do : = A
n
A
nn
A
n fgg limlimlim
Vy, lim lim .n nA
f d f d
Ch :
( )i Nu ggfn , kh tch trn A th b Fatou vn cn ng.
Tht vy, do gfn nn 0 gfn trn A.
T kt qu trn ta c:
( ) ( ) A
n
A
n gfgf limlim
V A
g hu hn nn :
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Hunh Vit Khnh SP. Ton 01-K.30 - 29 -
( ) ( ) ++AA
n
A A
n ggfggf limlim
hay, A
n
A
n ff limlim .
( )ii Nu ggfn , kh tch trn A th:
A
n
A
n ff limlim .
Tht vy,
Do gfn nn gfn v do hm g kh tch nn tho cu a. ta c:
( ) ( ) A
n
A
n ff limlim
( ) A nA n ff limlim Vy,
A
n
A
n ff limlim
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Hunh Vit Khnh SP. Ton 01-K.30 - 30 -
Chng 2: S HI T CA DY HM O
C
2.1 CC DNG HI T CA DY HM O C
2.1.1Hi t hu khp ni (converges almost everywhere)
nh ngha
Cho dy { }n nf v hm f o c trn A .
Dy { }n nf c gi l hi t hu khp ni v hm f trnA nu:
:B A ( ) 0B = v ( ) ( )lim nn
f x f x
= , \Bx A
K hiu: .a enf f hay. . .h k nnf f
V d:Xt cc hm s:
1 2,
, 3nn n
f n
= v 0f = trn [ ]0,1
Khi .a enf f trn [ ]0,1 .
Tnh cht
Cho { } gff nn ,, l nhng hm o c trn A . Khi , nu ffea
n . trn
A v .a enf g trn A th gf = h.k.n trn .A
Chng minh
V ff ean . trn A nn ( ) ( ) ( ): 0, lim , \Bn
nB A B f x f x x A
= =
V .a enf g trn A nn ( ) ( ) ( ): 0, lim , \Cnn
C A C f x g x x A
= =
Gi CBD =
Ta c: ( ) 0=D , v D\Ax th ( ) ( )xgxf =
Vy, gf = h.k.n trn .A
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Nh vy, nu ng nht cc hm s tng ng th gii hn h.k.n ca mt
dy nhng hm so c l duy nht.
nh l hi t b chn ca Lebesgue
Nu , ,nf g n hm g kh tch, v.a e
nf f (hoc trong o) trn A th:lim .nn
A A
f f
=
2.1.2Hi tu (converges uniformly)
nh ngha
Cho dy { }n nf v hm f o c trn A .
Dy { }n nf c gi l hi tu v hm f trn A nu:
( ) ( ) ( )0 0 00, : , nn n n n x A f x f x > = <
K hiu: nf fI .
V d
Trn ( )( ), , P vi l o m.
Xt dy hm ( ) 1 , 1
0 ,n
x nf x xx n
= >
Ta c ( )nf x I1
xtrn .
Tht vy, 0 01
0, , ,n n n x
> > > ta lun c:
( )0
1 1nf x
x n
< .
Vi { } ,n nf f l nhng hm o c trn A .
t ( ) { }n kk n
T r f f r
=
= >U
Khi ( )nT r l tp o c.
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Sau y l iu kin cn v mt dy hm o c hi tu.
nh l
Cho { } ,n nf f l nhng hm o c trn tp .A
Khi nf fI ( ) ( ) ( )0, : n rr n r T r > = .
Chng minh
( ) Gi s nf fI trn A
Khi :
( ) ( )0, : ,r n r k n r x A > th ( ) ( )kf x f x r
( ) ( )n rT r = .
( ) Gi s ta c ( )0, :r n r > ( ) ( )n rT r =
( ) ,k n r x A th ( ) ( )kf x f x r
Vy nf fI trnA .
2.1.3 Hi t u hu khp ni (converges uniformly almost
gfhfdheverywhere)
Cho dy { }n nf v hm f o c trn A .
Dy { }n nf c gi l hi tu hu khp ni v hm f trnA nu:
( ): 0B A B = v nf fI trn \B.A
K hiu: nf fI hu khp ni (hoc nf fI a.e).
2.1.4Hi t theo o (converges in measure)
nh ngha
Cho dy { }n nf v hm f o c trn A .
Dy { }n nf c gi l hi t theo o v hm f trnA nu:
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Hunh Vit Khnh SP. Ton 01-K.30 - 33 -
( ) ( ){ }0, lim : 0nn
x A f x f x
> =
K hiu: nf f trn A .
Nh vy,
nf f ( ) ( ){ }0 00, 0, 0 : , : nn n n x A f x f x > > > < .
V d: Xt cc hm s:
1 2,
n
n n
f
= , vi 2n v 0f = trn [ ]0,1
Khi nf f trm [ ]0.1 , vi l o Lebesgue trn .
Tnh cht ca dy hmhi t theo o
Cho l o.( )i Nu nf f
trn A , l o , v f g= h.k.n th .nf g
( )ii Nu nf f v nf g
th f g= h.k.n.
Chng minh
( )i Ta c: n nf g f f f g +
{ } { } { }0, 0n nf g f f f g > >
{ } { } { }0 0n nf g f f f g + >
V f g= h.k.n nn: { }0 0f g > =
Do , { } { }0 0n nf g f f khi n
Vy, .nf g
( )ii Ta c: n nf g f f f g +
Do vy, v
i0,
>
2
2
n
n
f ff g
f g
Do ,
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Hunh Vit Khnh SP. Ton 01-K.30 - 34 -
{ }2 2n n
f g f f f g
Suy ra,
{ } 2 2n nf g f f f g
+
Khi n th:
2nf f
0
2nf g
0
Suy ra { } 0f g = khi n
c bit,1
, 0n f gn
=
Mt khc,
{ } { }1
10
n
f g f g f gn
= > =
U
Suy ra { }1
10
n
f g f gn
=
Vy, f g= h.k.n.
2.1.5Hi t trung bnh (converges in the mean)
nh ngha
Dy { }n nf cc hm kh tch c gi l hi t trung bnh v hm kh tch f
trn A nu:
( ), 0n nA
f f f f d = khi n .
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Hunh Vit Khnh SP. Ton 01-K.30 - 35 -
nh l hi t trung bnh (Mean convergence theorem)
Gi s { }n nf , f l nhng hm o c trn A tha mn:
( ) ( )lim ,nn
f x f x x A
= , v tn ti nhng hm kh tch ,h g tha mn:
( ) ( ) ( ) nAxxhxfxg n ,, ( )1
Khi , nf f kh tch v nf hi t trung bnh v f .
Chng minh
Do { }n nf , f l nhng hm o c nn nf f cng o c.
T ( )1 cho n ta c: ( ) ( ) ( ) ,g x f x h x x A
( ) nh g f f h g
0 nf f h g , vi h g l hm kh tch.
Kt hp vi iu kin ( ) ( )lim ,nn
f x f x x A
= , p dng nh l hi t b chn
ca Lebesgue, ta suy ra:
lim 0nn
A
f f
=
Vy, nf hi t trung bnh v f .
2.1.6Hi t hu nhu (converges almost uniformly)
nh ngha
Dy hm o c { }n nf c gi l hi t hu nhu v hm o c f
trnA nu:
0, > ( ):B A B < v nf fI trn \BA
K hiu: .a unf f .
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Hunh Vit Khnh SP. Ton 01-K.30 - 36 -
nh l
Cho dy { }n nf v hm f o c trn A . Khi :
( )( ) 0. rTff nua
n khi n .
Chng minh
( ) 0 > cho trc, B o c: ( )B < , v nf fI trn \A E
Do , vi 0r> , ta c:
( ) ( ) ( ) ( ) ( )( ): n r n r n r T r B T r <
Do 0 > l ty nn( ) ( )( ) 0n rT r khi n .
( ) Do ( )( ) 0nT r khi n nn:
0 > cho trc, v vi mi s t nhin p sao cho:
1:
2pp n pn T
p
= ( ) ( ) 0, ,n mf x f x n m n < .
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Hunh Vit Khnh SP. Ton 01-K.30 - 37 -
2.2.2 Dycbn u ( uniformly Cauchy)
Dy hm o c { }n nf trn A c gi l cbn u trn A nu:
( ) ( )0 00, : , , .m nn m n n x A f x f x > <
2.2.3 Dycbn hu nhu (almost uniformly Cauchy)
Dy hm o c { }n nf trn A c gi l cbn hu nh u trn A nu:
( )0, :E E > < v { }n nf cbn u trn \E .A
2.2.4 Dy hm cbn trung bnh (Cauchy in the mean hoc meanfundamental)
Dy { }n nf cc hm kh tch c gi l cbn trung bnh nu:
( ), 0n m n mA
f f f f d = khi ,m n .
2.2.5Dy cbn trong o (Cauchy in measure, hoc fundamental
in measure).
Cho khng gian o ( ), , ,X A F F .
Cho dy { }n nf o c trn A vi l o.
Dy { }n nf c gi l cbn theo o trnA nu:
{ }0, 0n mf f > khi , .m n
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2.3 SLIN H GIA CC DNG HI T CA DY HM
O C
2.3.1 Lin h giahi t trung bnh vhi t theo o
nh l
Cho dy hm { }n nf , v hm f kh tch trn A . Khi nu nf hi t trung
bnh v f th ffn .
Chng minh
Vi s dng ty 0 > cho trc, t:
{ }n AB f f =
Ta lun c: ( )n nA B
f f d f f d B
T gi thit nf ht t trung bnh v hm f , ta c: 0nA
f f d khi n
Do , ( ) 0B khi n
Nh vy, nf f .
Chiu ngc li ca nh l ni chung khng ng, tc l, nu dy hm
nf f , th ta cha th suy ra c nf ht t trung bnh v hm f . Sau y l
mt v d minh ha.
V d:
Xt dy hm nf trn [ ]1,0 c xc nh nh sau:
( )
1, 0,
10, ,1
n
n xn
f xx
n
=
v hm 0f = .
Ta c: [ ]{ }( )1 1
0,1 : 0; 0nx f f n n
= =
khi .n
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Hunh Vit Khnh SP. Ton 01-K.30 - 39 -
Do , .nf f
Tuy nhin, [ ]
11
0 0
1, 0,1n
nf f d nd x = =
nn nf khng hi t trung bnh v hm f .
2.3.2 Lin h giahi t trung bnh vhi t hu khp ni
nh l
Cho dy hm { }n nf kh tch trn A . Ngoi ra,.a e
nf f . Khi nu tn ti
hm kh tch khng m g sao cho nf g th nf hi t trung bnh v hm f .Chng minh
Do .a enf f nn. 0a enf f
Ta cn c: n nf f f f + 2g
p dng nh l hi t b chn ta c:
0nA
f f khi n
Vy, nf hi t trung bnh v hm f .
V d sau cho thy { }n nf hi t hu khp ni v hm f , nhng { }n nf
dfghdfkhng hift trung bnh v hm f .
Xt ( ) 2 21nn
f xn x
=+
.
Ta c: [ ]1,0x ,
( ) 2 22
1lim lim lim 011nn n nnf x n x nxn
= = =+ +
Do , . 0a enf trn on [ ]1,0 .
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Hunh Vit Khnh SP. Ton 01-K.30 - 40 -
Tuy nhin, ( )1 1
2 20 0
lim lim1nn n
nf x dx dx
n x =
+
[ ] ( )020
1lim lim lim
1 2
nu nxn
n n n
du arctgu arctgnu
=
= = = =
+
Nh vy, nf khng hi t trung bnh v 0.
nh l
Cho dy hm { }n nf , v fkh tch.
Nu nf hi t trung bnh v hm f th tn ti dy con { } { }kn nf f sao cho
.
.ka e
nf f Chng minh:
Do nf hi t trung bnh v hm f nn nf f
Suy ra tn ti dy con { } { }kn n
f f sao cho . .k
a enf f
2.3.3 Lin h giahi t theo o vhi t hu khp ni
nh l
Cho dy hm { }n nf o c trn A , hi t hu khp ni v f trn A v
l o th f o c trn A .
Nu ( )A < th .nf f
Chng minh
Ta chng minh f o c trn A .t { }: nB x A f f = / , khi ( ) 0B =
Do nn f o c trn B .
Mt khc, trn \B,A nf f
f o c trn \B.A
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Hunh Vit Khnh SP. Ton 01-K.30 - 41 -
Nh vy, fo c trn ( )\BA B A= .
Gi s ta c ( )A < , ta cn chng minh nf f trn .A
Vi 0 > cho trc, ta c:
Nu , : n in i f f + th x B
{ }1 1
n in i
f f B+
I U
Do nn { }1 1
0n in i
f f +
=
I U
t { }1
n n ii
E f f +
= U
Nh vy, ta c:1
0nn
E
= I
Khi n tng, s hng n if + gim. Do , s phn t ca A trong nE b t i. V
th, ta c 1 2 3 ... ...nE E E E
Mt khc, ( ) ( )1E A < . Do ta c ( )lim 0nn E =
( )1lim 0nnE
=
Ta c:
{ } { } { }1 1 11 2
n n i n n ii i
E f f f f f f + +
= =
U U
Ta c: { }( ) ( )10 n nf f E
{ }( )lim 0nn
f f
=
Vy, .nf f
C nhng dy hm hi t hu khp ni nhng khng hi t hu theo o.
V d
Xt tp hp s thc vi o Lebesgue.
Chn[ ], 1n n n
f += .
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Hunh Vit Khnh SP. Ton 01-K.30 - 42 -
Ta c 0nf khi n
Tuy nhin, nf khng hi t hu theo o v 0 v vi1
2 = th:
( ) [ ]1
0 , 1 1 02x f x n n
= + = .
nh l
Cho dy hm { }n nf , v hm f o c trn A . Khi , nu nf f trn
A th tn ti dy con { } .: .k k
a en nn
f f f
Chng minh
Chn dy s dng { } : 0,k kk v dy { }1
: 0, .k k kkk
=
> <
Do nf f nn ( ) ( ) { }, : ,k n k k n k n n k f f
t ( ) ( ){ }1 2 11 , max 1, 2 ,...n n n n n= = +
1 2 ...n n < < v { }( ), kn k kk f f
t { },ki n kk iQ f f
== U v
1i
iB Q
== I
( ) ( ) { }( ) 0ki n k k k i k i
B Q f f
= =
( ) 0B =
Mt khc, \Bx A : ii x Q
, 0kn k
k i f f khi k
Vy . .k
a enf f
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2.3.4 Lin h giahi t trung bnh v hi tu
nh l
Nu { }n nf l dy hm kh tch hi tu v hm f trn A , v ( )A < + ,
th nf hi t trung bnh v hm f .
Chng minh
Do nf fI nn: ( ) ( )0 00, : , ,nn f x f x n n x A > <
Ta c n nf f f f +
nn n nA A A
f f f f +
( ). A + nA
f < +
Vy, f kh tch trn A .
Mt khc, 0 ,n n ta c:
( ) , 0nA
f f A >
0, .nA
f f n
Vy, nf hi t trung bnh v hm f .
2.3.5 Lin h giahi t hu nhu vhi t hu khp ni
nh l
Cho { } ,n nf f l nhng hm o c trn A . Khi nu.a u
nf f trn A
th .a enf f trn .A
Chng minh
Do .a unf f nn vi mi s t nhin k, tn ti kE sao cho ( )1
kE k < v
nf fI trn .ckE
-
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 44 -
t1
ck
k
F E
=
= U th nf f trn F
Mt khc, ( ) ( )1
,c kF E kk <
( ) 0cF =
Vy, .a enf f trn A .
Chiu ngc li ca nh l ni chung khng ng. Tuy nhin nu c thm
mt siu kin, th chiu ngc li sng. C th, ta c nh l sau:
nh l
Cho { } ,n nf f l nhng hm o c trn A , ( )A < + . Khi nu
.a enf f th
.a unf f .
Chng minh
Vi ,k n l nhng s nguyn dng cho trc, t:
1kn m
m n
E f fk
=
= cho trc, ( )kn: \E ,2k kkn A n n
<
t1
k
kn
k
F E
=
= I , th Fo c, v:
( ) ( ) ( )k
kn n
11
\F \E \Ek
kk
A A A
==
=
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Hunh Vit Khnh SP. Ton 01-K.30 - 45 -
2.3.6 Lin h giahi t theo o vhi t hu nhu
nh l
Cho dy hm { }n nf , f l nhng hm o c trn A . Khi , nu
.a unf f th nf f
.
Chng minh
Do .a unf f nn vi mi s t nhin m , tn ti tp o c mE tha:
( )1
mE m < , v nf fI trn m\EA
( ) ( ), : ,N m n N m > th { }n mf f E { }( ) ( )
1,n mf f E mm
<
{ }( ) 0nf f =
Vy, nf f .
Chiu ngc li ca nh l ny ni chung khng ng. Tuy nhin, ta c
nh l sau:
nh l
Cho dy hm { }n nf , f l nhng hm o c trn A .
Nu nf f , th tn ti dy con { }
kn kf { }n nf sao cho
.
k
a unf f .
Chng minh
Do nf f nn ta c:
k , t ( ) ( )1
n nE x f x f xk
=
th ( ) 0nE khi n
Chn kn sao cho kn n > ta c ( ) 2k
nE<
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Hunh Vit Khnh SP. Ton 01-K.30 - 46 -
t ( ) ( )1
kk nA x f x f x
k
=
v j k
k j
G A
=
= U
Khi , vi C Cj kx G x A , vi k j
Suy ra, ( ) ( ) 1knf x f x k < , vik j
t jG G= , ta c: ( ) ( ) ( )12 2 ,k jj k
k j k j
G G A j
+
= =
= =
Suy ra, ( ) 0G = .
Nu ,jx G x G j ( ) ( )1
knf x f x
k < , k
Vi 0 > , chn k sao cho1k < . Khi ta c:
,x G l kn n > ( ) ( )1
lnf x f x
k < <
Do kn khng ph thuc vo x nn ta c.
k
a unf f .
nh l Riesz
Nu { }n nf l dy hm o c l cbn theo o trn ( ),A A < + , th tnti dy con { }
kn kf , v hm o c f sao cho
knf hu khp ni, hu nhu, v
theo o v f trn .A
Chng minh
Do dy hm o c { }n nf l cbn trong o, do :
( ) ( ) { }( )0, : , m nN m n N f f >
t ;21
1
=Nn
1
1max 1, , 1
2k k kn n N k +
= +
Xt dyknk
fg = l dy con ca { }nnf , dy hm { }kkg c tnh cht sau:
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Hunh Vit Khnh SP. Ton 01-K.30 - 47 -
Nu t
= + kkkk ggE 2
11 th ( )
1
2k kE
t I U
=
=
=1n nk
kEF th ( ) 0=F
Mt khc, \ , : \ ,x k xx A F k x A E k k
Do vy, kij > , ta c:
( ) ( ) ( ) ( ) ( ) ( )xgxgxgxgxgxg iijjij ++ + 11 ...
1
1 1...
2 2j i< + +
1
1
2i<
( )xgk l dy cbn trong , \R x A F
( )xgk l dy hi t
t:
( )( )lim , \
0 , \
kk
g x x A F f x
x A F
=
Nh vy, fg eak . trn .A
Vi 0> cho trc, v s t nhin k tha mn1
1
2k .
t: U
=
=kj
kk EF
Khi : ( ) th:
( ) ( ) ( ) ( ) ( ) ( )xgxgxgxgxgxg iijjij ++ + 11 ...
ij 2
1...
2
11
++
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 48 -
12
1 i
Cho j ta c: ( ) ( ) 1 11 1
, ,2 2
Ci Ki k
f x g x x F i k <
Suy ra, ig hi tu v f trn ckF
Vy, fg uai . trn .A
Mi dy hi thu nhu th hi t hu khp ni. Do , ta c th chon
dy con { }iig ta c .fgi
2.3.8 Lin h giahi t hu khp ni vhi tu
nh l Egoroff
Cho ( ) , tn ti ( ): CE A E < , v nf hi tu v hm f trn E.
Chng minh
Do .a enf f trn A nn:
C A : ( ) 0C = v ( ) ( )lim nn
f x f x
= \x B A C =
Vi mi ,m n , t:
( ) ( ),1
:m n jj n
B x B f x f xm
=
=
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Hunh Vit Khnh SP. Ton 01-K.30 - 49 -
t ,1
\mm n
m
D A B
=
= I
Ta c: ( ) ( ),1 1
\2mm n mm m
D B B
= =
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 50 -
2.3.10 Lin h giacbn trung bnh vcbn theo o
Nu dy { }n nf cc hm kh tch l c bn trung bnh th { }n nf l c bn
trong o.
Chng minh
Vi s dng ty 0 > cho trc, t:
{ }mn n m AA f f =
Ta lun c: ( )mn
n m n m mn
A A
f f d f f d A >
Do { }n nf l dy cbn trung bnh, suy ra 0n mA
f f d khi ,n m
( ) 0mnA khi , .n m
Nh vy, { }n nf l cbn trong o.
2.3.11 Lin h giacbn trung bnh vhi t hu nhu
nh l
Cho { }n n
f l nhng hm kh tch trn A . Khi nu { }n n
f l cbn trung
bnh trn A th tn ti dy con { }kn k
f ca dy { }n nf , v hm o c f trn A sao
cho .k
a unf f trn A .
Chng minh
V dy { }n nf l cbn trung bnh trn nn { }n nf l cbn trong o.
Theo nh l Riesz, tn ti dy con { }kn k
f ca { }n nf , v hm f sao cho
.k
a unf f trn A .
2.3.12 Lin h giacbn hu nhu vhi t hu nhu
nh l
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Hunh Vit Khnh SP. Ton 01-K.30 - 51 -
a. Dy hm o c { }n nf tha.a u
nf f trn A th { }n nf l dy cbn hu
nhu trn .A
b. Nu { }n nf l cbn hu nhu trn A th tn ti hm o c f sao cho
.a unf f
Chng minh
a. Do .a unf f nn ( )0, :B A B > < , v nf fI trn \B.A
Khi ,
( ) ( )0 00, : , \B2nn n n f x f x x A
> > <
Nh vy, 0 00, : ,n n m n > > , ta c:
( ) ( ) ( ) ( ) ( ) ( ) , \Bm n m nf x f x f x f x f x f x x A + <
{ }n nf l cbn u trn \BA .
Vy, { }n nf l cbn hu nh u trn .A
b. Ly1
, 1, 2,...kk
= =
Khi , kE : ( )1
kEk
< v { }nn
f l cbn u trn \ kA E
t1
kk
E E
=
= I
Ta c: ( )1
,E kk
<
Do : ( ) 0E =
Nu :c Ckx E k x E
( ){ }n nf x l dy cbn
Do , tn ti ( )f x : ( ) ( )lim nn
f x f x
=
Ta xem ( ) 0f x = trn E
Ta c: .a enf f
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 52 -
Vi 0, 0, ,F n > sao cho:
( )F < v ( ) ( ) 0, , ,C
n mf x f x x F m n n < >
t CG F E= ta c ( )G <
Cho m ta c ( ) ( ) 0, ,nf x f x x G n n <
Do nf fI trnCG
Nh vy .a unf f .
2.3.13 Lin h giacbn theo o vcbn hu nhu
Cho dy hm { }nnf o c trn A . Khi , nu { }nnf l cbn theo o
th tn ti dy conknk
f sao choknk
f l cbn hu nhu.
Chng minh
Do { }nnf l cbn trong o trn A nn , :kk m
1 1, ,
2 2m n kk kf f m n m
<
( )1
t: ;11 mn =
{ };,1max 212 mnn += { };,,1max 323 mnn +=
Khi , dy s .....,, 21 nn s xc inh cho ta dy con knkf ca { }nnf
Vi mi k, t:
1
1
2k kk n n kE f f
+
=
,i kk i
B E i
=
= U
Ta chng minhkn
f l cbn u trn ciB .
Vi mi i v s thc 0> bt k, ta chn sr, sao cho ,isr v 11
2s <
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 53 -
Vi mi Cix B , ta c:
1r s k k n n n nk s
f f f f +
=
12kk s
=
121
s
<
Nh vy, dy conknk
f l cbn u trn ciB
Mt khc, 0> ,1
1:2i
i <
Kt hp vi ( )1 ta c:
( ) ( ) 11 1
2 2i k k ik i k iB E
= =
= cho trc, ta lun c:
m n m nf f f f f f +
{ }2 2m n m n
f f f f f f
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 54 -
{ }( )2 2m n m n
f f f f f f
+
Cho ,m n , kt hp vi iu kin nf f trn A , ta c:
{ }( )m nf f 0
Vy, { }n nf l dy cbn trong o.
( ) Gi s{ }n nf l cbn trong o.
Khi , tn ti dy con l { }kn k
f sao cho { }kn k
f l dy cbn hu nh
u.
Gi ( ) ( )limknk
f x f x
= , vi x m ti gii hn tn ti.
Ta c, vi mi 0 > :
{ }2 2k kn n n n
f f f f f f
Suy ra:
{ }( ) 0nf f khi n
Vy, .nf f
2.3.15 Lc th hin mi lin h gia cc dng hi t.
Nh vy, s hi t ca cc dy hm o c c mi quan h vi nhau. d
nm c cc mi quan h ny, sau y l lc th hin mi quan h gia chng.
Tuy nhin, mi quan h ny c s thay i, ty thuc c hay khng iu kin
( )A < . Sau y l hai trng hp ny:
K hiu: uni: hi tu (uniformly).au: hi t hu nh u (almost uniformly).
a.e: hi t hu khp ni (almost everywhere).
meas: hi t trong o (in measure).
mean: hi t trung bnh (in mean).
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 55 -
Trng hp ( )A <
Trng hp tng qut
a.e
mean
a.u
meas
uni
mean
a.u
meas a.e
uni
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 56 -
Chng 3: BI TP
Bi gii
Ta nh ngha hm nf nh sau:
( )
, tn ti s 00
1:n x
n<
Khi : 0n n > ta c:1
xn
<
( ) 0nf x = < .
Tuy nhin, ta c:
[ ] 011,0
1,0 /== nn nddf khi n
Vy, { }nnf khng hi t trung bnh v .f
Bi gii
Trn khng gian o [ ]1,0=X vi o Lebesgue trn r , t:
( ) nn xxf = v ( ) 0=xf .
( )1 Chng minh ffn hu nhdu.
Bi 1: Cho v d v dy hm { } [ ]1,01Lf nn sao cho.a e
nf f nhng { }nnf
khng hi t trung bnh v .f
Bi 2: Cho v d v dy hm o c tha mn cc iu kin sau:
( )1 Hi thu nhu.
( )2 Khng hi tu.
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 57 -
Vi 0> , t{ }
2
1,min = , v [ ]1,1 =A
Ta c:
( )m A = < , ( ) ( ) ( )n
n xfxf 1 , \Ax X
Vy, nf hi t hu nhu trn X.
( )2 Chng minh nf khng hi tu v f trn X .
Ta thy ti 1=x th ( ) 1xfn nn ffn /
Vy, nf khng hi tu v hm f trn [ ]0,1 .
Bi gii
( ) V nf f nn:
( ) ( ){ }0, 0, , : : nN n N x f x f x > > <
Chn = ta c:
( ) ( ){ }0, , : : nN n N x f x f x > <
( ) 0, 0, :mm
> > <
Vi 0m
> theo gi thit ta c:
, :N n N ( ) ( ): nx f x f xm m
<
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 58 -
Bi giiTa c: nf f
trn ( ) ( ){ }0, lim : 0nn
x f x f x
> =
( ) ( ){ }0, , : : nN n N x f x f x > =
( ) ( )0, , , : nN n N x f x f x >
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 59 -
T ( )1 v ( )2 ta c:
=AA
nn
ff .lim
Bi gii
( )a V lin tc nn:
( ) ( ){ } ( ) ( ){ }: :n nx f x f x x f x f x / /o o
( ) ( ){ }( ) ( ) ( ){ }( ): :n nx f x f x x f x f x / /o o
T gi thit: ff ean . suy ra:
( ) ( ){ }( ): nx f x f x / 0=
( ) ( ){ }( ): nx f x f x /o o 0=
Vy, . .a enf f o o
( )b Do l lin tc u nn:
( ) ( ) ( ) ( ) .:0,0 yxyx
Trng hp ffn u.
Do nf hi tu v hm f nn:
( ) ( ) ( ) ( ) > xfxfAxnnn n,:,0 00
Do , ( )( ) ( )( ) 0, ,nf x f x n n x A < > o o .
Vy, nf f o o u trn A .
Trng hp ffn hu nhu.
Bi 6: Cho dy hm { }nn
f , v hm f nhng hm o c trn A, v hm
: . Chng minh rng:
( )a Nu lin tc v ff ean . th . .a enf f o o
( )b Nu lin tc u v ffn u, hu nhu, hay theo o th
ffn .. u, hu u, hay theo o.
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 60 -
Do ffn hu nh u trn A nn: Vi 01 > , E : ( ) 1 > < <
Do nu:
( ) ( ) ( ) ( )n nf x f x f x f x < cho trc, t:
{ }= ffE nn
Khi :
Bi 7: Cho ( ),X l khng gian o tha mn ( )
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Hunh Vit Khnh SP. Ton 01-K.30 - 61 -
( ) +
++
=
nn EX n
n
E n
nn ff
ff
ff
ffffd
\ 11,
( ) ( )\n nE X E +
Do l s dng ty , v ( ) nn 0 > , ta c:
( ) ffd n , +
nE n
n
ff
ff
1( )
1 nE
+
V ( ) 0, ffd n nn ( )nE 0
Vy, nf f
Bi gii
Cch 1:
V dy hi tu v hm f nn N nguyn dng tha:
1,nf f n N <
Nh vy, 1+< Nff
Hm 1+Nf l hm kh tch v Nf l hm kh tch v ( ) ( )
0 0: ,nn f f x X n nX
< >
Mt khc, 0nn , ta c :
Bi 8: Gi s ( ),, FX l mt khng gian o hu hn . Gi thit { }nf l dy
hm kh tch hi tu v hm f . Chng minh rng:
=XX
nn
fddf lim
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 62 -
( )( )
n n n
X X X X X
f d fd f f d f f d d A
= < =
Nh vy, lim .nn
X X
f d f d
=
Bi gii
( )a Ta bit, ( )xfnn lim tn ti khi v ch khi :
( ) ( )lim limn nf x f x= .
Nh vy, L = ( ) ( ){ }:lim limn nx f x f x=
Ta c nflim v nflim l nhng hm o c v nf o c .
t nn fff limlim = Ta c f l hm o c
Do , { }( )01= fL o c.
( )b Do ffea
n
.
nn ( ) 0: = CEFE v ffn trn E
Vi Ra , ta c ( ){ } ( ){ } ( ){ }axfExaxfExaxfXx C >>=> :::
( ){ } ( ){ }axfExaxfEx Cn >>= :lim:
Tp th nht v phi l tp o c , tp th hai l tp con ca CE l o
c ( do )
Do ( ){ }axfXx > : l o c
Suy ra f o c.
Bi 9: Cho { }nf l dy hm o c . Chng minh rng :
( )a L = ( ){ }:lim nn
x X f x tn tai
o c.
( )b Nu ( ),, FX , vi l o , v .a enf f th f o c.
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Hunh Vit Khnh SP. Ton 01-K.30 - 63 -
Bi gii
,0> t ( ){ }.: = xfxA nn Ta c : ( )0n
n n n
A A
A f d f d
V 0 dfA
n nn ( ) 0nA khi n
Vy, .0nf
Bi gii
V ffn nn { }nn ff k tha ff
eank
. trn .
Do ,peap
n ffk .
trn .
p dng b Fatou, ta c,
( )lim lim 1k kp p p
n n
R R R
f d f d f d =
Vy, .Bf
Bi gii
t n nA
a f d= v naa lim=
Khi , tn ti dy aaakk nn
: khi k
Bi 10: Cho dy hm o c khng m { }nf trn A tha mn: .0lim = dfA
n
Chng minh rng: .0nf
Bi 11: Vi 0>p , t =B { RRf : tha fo c v 1 dfp
R}.
Gi s rng { } Bf nn v .ffn Chng minh rng: .Bf
Bi 12: Cho dy hm o c { }nf ,f l nhng hm kh tch Lebesgue. Chng
minh rng: nu ffn trn A th .lim dfdf
A
n
A
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LUN VN TT NGHIP
Hunh Vit Khnh SP. Ton 01-K.30 - 64 -
Do ffn nn
knf f khi k
V nh vy, tn ti dy con { } { } .:k k km ma e
n n nkmf f f f
Theo b Fatou, ta c:
lim lim limk k km m m
n n nm m m
A A A
f d f d f d a a
= = =
Nh vy, .lim dfdfA
n
A
Bi gii
Bi v ( ) ( )xgxfn hu khp ni v ( )g x kh tch trn nn nf kh tch trn .
p dng b Fatou, ta c:
lim nR R R
f d f d gd <
Suy ra f kh tch Lebesgue trn .
Vy, theo nh l hi t b chn, ta c:
lim nn
R R
f d f d
=
Do vy, 0lim = dffR
nn
Vi 20 00, : nR
n f f d n n > < >
t ( ) ( ){ }>= xfxfRxB n: th ( )21
n
R
B f f d
< < =
t \A B= .
Bi 13: Cho nff, l cc hm o c trn tha.a e
nf f trn R v tn ti
hm g kh tch trn R tha ( ) ( )xgxfn hu khp ni. Chng minh rng ffn hu nhu.
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Hunh Vit Khnh SP. Ton 01-K.30 - 65 -
Trn A ta c ( ) ( ) 0nnxfxfn >
Suy ra ffn hu nhu trn .
Bi gii
Do ffn nn { } { } :
k kn n nnkf f f f khi k
V knf g , g l hm kh tch, nn theo nh l hi t b chn, ta c:lim
knkf f
= ( )
M lim limkn nk k
f f
= , kt hp vi ( ) ta c: lim nn f f = .
Bi gii
Theo nh ngha ca lim nf ta c: { } { }kn n nkf f : l im limkn nk f f =
Vkn
f f nn .:k k kj j
a en n nf f f f
Theo b Fatou ta c: lim limk kj j
n nf f f = = l im knk f = lim nf
Vy, lim nf f
Bi 16: Cho ( ) ,nn E
E < hi t trung bnh v hm f . Chng minh Ef = h.k.n
vi E l mt tp o c.
Bi 15: Cho 0nf v nf f . Chng minh rng:
limf f
Bi 14: Cho nf g , g l hm kh tch v nf f . Chng minh rng:
lim nf f=
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Hunh Vit Khnh SP. Ton 01-K.30 - 66 -
Bi gii
t
=
=j
njnEE
1U ta c E o c v EEn khi n
DonE
hi t trung bnh v hm f nn fnE
. T tn ti dy con
feaEkn
. .
Vy Ef = h.k.n.
Bi gii
a. Vi ,0> ta lun c:
( ) ( ){ } : :2 2n n n n
f g f g x f f x g g
+ +
( ) ( ){ } : :2 2n n n n
f g f g x f f x g g
+ + +
Cho n , ta c:
( ) ( ){ } 0n nf g f g + +
Vy, gfgf nn ++ .
b. Ta c:
( )i Nu nf f th, naf af
, a .
( )ii Nu nf f th 2 0nf
iu ny suy ra t{ } { }.2 = nn ff
( )iii .gfgfn
Bi 17: Cho { }nnf , { }nng l cc dy hm o c trn A , ffn , ggn
.
Chng minh rng:
( )a gfgf nn ++ .
( )b gfgf nn ..
nu ( ) +
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Hunh Vit Khnh SP. Ton 01-K.30 - 67 -
Chng minh:
Ta c: { }=>In
ng .
Do ( )A < nn { }( ) 0g n >
Vi 0, > v 0 > tn ti 1n sao cho { }( ) 21
ng , chn 0n sao cho:
01 2
nn n f f n
=+22
Vy, .gfgfn
( )iv Ta chng minh ffn th 22 ffn
.
V ffn
nn .0
ffn
Do , theo( )ii ta c:
( )2
0nf f
Theo ( )i , ( )iii , v ( )a ta c:
( )22 2 2 2 0n n nf f f f f f f
= +
Vy, 22 ffn
( )v Sau cng, p dng cc bc trn v kt qu t cu a, ta c:
( ) ( )2 22 2 2 21 1 1 1 1 1
2 2 2 2 2 2n n n n n nf g f g f g f g f g fg= + + =
Vy gfgf nn .
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Hunh Vit Khnh SP. Ton 01-K.30 - 68 -
Bi gii
( ) ( )iii do nh ngha hi t theo o.
( )ii ( )iii do { } { }. > ffff nn
( ) ( )iiiii ,0,0 >> nu
,
2
Nn th :
{ }( ) ffn
>
2
ffn < .
( ) ( )ivii Khi . =
( ) ( )viv Do { } { }. > ffff nn
( ) ( )iiv ,0,0 >> chon
=
,
2min .
Khi , nu n P th:
{ }( ) ffn
> 2
ffn { }( )nf f > .
Bi 18: Cho ffn , l nhng hm o c trn A . Chng minh rng cc mnh
sau l tng ng.
( )i .ffn
( )ii ,:,0,0 NnN >> th { }( ) . ffn
( )iii ,0, 0, :N n N > > th { }( ) . > ffn
( )iv PnP > :,0 th { }( ) . ffn
( )v 0, :P n P > th { }( ) . > ffn
Bi 19: Gi { }nnE l dy nhng tp con o c ca [ ]ba, . Chng minh rng:
( )a Chng minh rngnE
hi tu v 0 trn [ ]ba, nu v ch nu =nE
vi n ln.
( )b Chng minh rng 0nE
trn [ ]ba, nu v ch nu ( ) = nn
Elim 0.
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Hunh Vit Khnh SP. Ton 01-K.30 - 69 -
Bi gii
( )a ( ) Vi ,2
1= chn 0N sao cho 0Nn > th 21 .
( ) Nu = nEN :0 , 0Nn > th ( ) 0=xnE . Do 00 0,nE n N = >
Vy,nE
hi tu v 0 trn [ ]ba, .
( )b Vi 10 = . Khi , ( )rBk o c v ( )( ) A
nk ffrrB
1
Vi ( ) ( )U=
=nk
kn rBrT , t gi thit ta c:
( )( )
=nk Akn ffr
rT1
0 khi 0n
Vy, ff uak . .
Bi gii
Chn 1 2,
.nn n
f n
=
Bi 20: Gi s{ } ff kk , l nhng hm kh tch trn A tha mn
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Hunh Vit Khnh SP. Ton 01-K.30 - 70 -
Vi 0 > , chn2
:N N
<
t2
0,AN
=
Ta c ( ) 2AN
= <
Mt khc, Cx A , ta c: 2 2 ,x n N N n
> >
Do , n N > v Cx A th ( ) 0nf x =
Vy, nf hi thu nhu v hm 0f = .
Tuy nhin,
1 2,
1. . 1n
n n
f d n nn
= = =
Do , 0nf f khi n
Vy, nf khng hi t trung bnh v f .
Bi gii
Do l o -hu hn nn tn ti cc tp , 1, 2,...jA j = sao cho:
1j
j
X A
=
= U , v ( ) ,jA j <
Khi , vi mi j , tn ti j jE A sao cho:
( )1
\j jA Ej
< , v nf fI trn jE
Ta c:
( )( )1 1
\ \j j j jj j
E A A E
= =
=U U
Bi 22: Cho l o -hu hn, .a enf f . Chng minh rng tn ti cc
tp 1 2, ,...E E trong X sao cho 1 0
C
jj E
=
= U vnf fI trn ,jE j
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Hunh Vit Khnh SP. Ton 01-K.30 - 71 -
( )( )1
\ \j jj
X A E
=
= U
( )1
\ \j jj
X A E
=
= I
Do :
( ) ( )1 1
1\ \
C
j j j j jj j
E A E A E j
= =
= <
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Hunh Vit Khnh SP. Ton 01-K.30 - 72 -
KT LUN
Nh vy, i vi dy hm o c c nhng dng hi khc nhau nh: hi t
theo o, hi t hu khp ni, hi t hu nhu, hi t trung bnh, V gia cc
dng hi t ny c mi lin h vi nhau. Cc mi lin h ny c s thay i khi
chng c xt trong khng gian o hu hn.
Tuy nhin, c th lun vn cha kp khai thc ht sa dng ca cc dng
hi t cng nh l mi lin h gia chng. Trong tng lai, nu c iu kin, em s
tip tc nghin cu su hn khai thc thm v ti ny.
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TI LIU THAM KHO
[1] Trn Th Thanh Thy, o v tch phn lebesgue, i hc Cn th,2007.
[2] u Th Cp,o v tch phn, NXB Gio dc, 2007.[3] Robert G. Bartle, A Modern Theory of Integration, American
Mathematical Society, 2001.[4] Robert G. Bartle, Solution Manual to A Modern Theory of Integration,
American Mathematical Society, 2001.[5] Shmuel Kantorovitz, Introduction to Modern Analysis, Oxford
Mathematics, 2004.[6] Avner Friedman, Foundations of Modern Analysis, Dover Publications,
2001.