View
38
Download
13
Category
Preview:
DESCRIPTION
Σημείωσεις του Μαθηματος Θεωριά Μέτρου του Τμήματος Μαθηματικών του ΕΚΠΑ
Citation preview
pi , 2014
ii
, pi pi .
1
1 - 51.1 - . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Dynkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 152.1 . . . . . . . . . . . . . . . . . . . . . . 152.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 293.1 Lebesgue . . . . . . . . . . . . . . . 29
3.1.1 Lebesgue . . . . . . . . . . . . . . . . . . 313.1.2 . . . . . . . . . . . . . . . . . 38
3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 . . . . . . . . . . . . . . . . . . . . . 433.4 pi . . . . . . . . . . . . . . . . 453.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Lebesgue 514.1 Lebesgue . . . . . . . . . . . . . . . . . . . 514.2 Lebesgue . . . . . . . . . . . . . . . . . . 564.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.4 pi Borel . . . . . . . . . . . . . . . . . . 64
4.4.1 Cantor . . . . . . . . . . . . . . . . . . . . . . 644.4.2 Cantor-Lebesgue . . . . . . . . . . . . . . . . . 68
4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 735.1 . . . . . . . . . . . . . . . . . . 735.2 . . . . . . . . . . . . . . . . 775.3 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.4 . . . . . . . . . . . . . . . . . . . . 855.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
iv
6 916.1 pi . . . . . . . . . . . . . . 926.2 . . . . . . . . . . . . . . . . . . 94
6.2.1 . . . . . . . . . . . . . . . . . . . . . 1026.3 . . . . . . . . . . . . . . . . . . . . . . . 1046.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7 1157.1 . . . . . . . . . . . . . . . . . 1157.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.4 . . . . . . . . . . . . . . . . . . . . . . 1257.5 . . . . . . . . . . . . . . . . 1277.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8 1358.1 pi . . . . . . . . . . . . . . . . . 1358.2 Luzin . . . . . . . . . . . . . . . . . . . . . . . . . . 1388.3 Riemann . . . . . . . . . . . . . . . . . 1408.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9 1459.1 . . . . . . . . . . . . . . . . . . . . . . . . 1459.2 Tonelli Fubini . . . . . . . . . . . . . . . . . . . . 1539.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10 Radon-Nikodym 16110.1 pi . . . . . . . . . . . . . . . . . . . . 16110.2 Lebesgue-Radon-Nikodym . . . . . . . . . . . . . . . . 16310.3 . . . . . . . . . . . . . . . . . . . . . 16710.4 Lebesgue . . . . . . . . . . . . . . . . . . . . 17110.5 pi Riesz . . . . . . . . . . . . . . . . . 17210.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
11 Lp 17711.1 Lp . . . . . . . . . . . . . . . . . . . . . . . . 17711.2 Lp . . . . . . . . . . . . . . . . . . . . . 18111.3 L1 L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18611.4 pi Radon-Nikodym . . . . . . . . 18911.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Riemann 199.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.2 Riemann . . . . . . . . . . . . . . . . . . . . . . . . 201.3 Riemann . . . . . . . . . . . . . . . . . 203.4 Riemann . . . . . . . . . . . . . . . . . . . . . . . . . 206
pi 209.1 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.2 pi . . . . . . . . . . . . . . . . . 211
( ) pi 20 pi . Riemannpi Lebesgue, pi, pi , pi pi pi.
pi R . Riemann f : [a, b] R, Riemann f pi pi f , b
a
f(x)dx = (Rf ), (1)
piRf =
{(x, y) R2 : x [a, b] 0 y f(x)}. (2)
a b
Rf
1:
Riemann pi :
1. pi
P = {a = x0 < x1 < x2 < ... < xn = b}
pi pi ,
mk = inf{f(x) : xk x xk+1} Mk = sup{f(x) : xk x xk+1}
2
pi
L(f, P ) =
n1k=0
mk(xk+1 xk) U(f, P ) =n1k=0
Mk(xk+1 xk).
2. pi P , pi L(f, P ) U(f, P ) pi .
3. , pi 0, pi , f Riemann f .
Lebesgue, pi pi , :
pi . , f 1 f([a, b]) [m,M ],
Q = {m = y0 < y1 < y2 < ... < yt = M}., pipi :
L(f,Q) =
t1k=0
yk`({x [a, b] : yk f(x) < yk+1})
U(f,Q) =
t1k=1
yk+1`({x [a, b] : yk f(x) < yk+1}),
pi `(A) A R. A ( ) pi pi . pipi , pi A pi pipi, pipi .
pi pi, pi Lebesguepipi pi pi , . pipi R pi, pi :. pi ` : P(X) [0,] I R, `(I) ( ) pi pipipi pi ;
pi pi pi: An pi R,
`
( n=1
An
)=
n=1
`(An). (3)
, pi `pi pi pipi pi pi pi pi pi pi -.
pipi Lebesgue pi :
1 pi pi pi .
3
1. pi ( pi -), pi ( ) . , pi R Rk.
2. f : Rk R pi pi pi pi Riemann .
3. R pi pi pi pi , .
4. pi Lebesgue pi pi pi Riemann.
: Riemann - pi . , Riemann fn : [a, b] R f fn f R,
fn(x) f(x), x [a, b]
pi pi ba
fn(x)dxf(x)dx.
pi, pi f Riemann.
. {qn : n = 1, 2, ...} [0, 1]. fn : [0, 1] R f : [0, 1] R
fn = {q1,q2,...,qn} 2 f = Q[0,1]
pi fn f (;) fn Riemann - ( pipi pi ), f ( pi Riemann, pi ).
5. , pi pi Lebesgue pi Riemann: Riemann Lebesgue .
pi pipi pi . pi 6 , Lebesgue. pipi 5 pi pi- .
2 A R, A(x) ={
1 , x A0 , x / A
1
-
pi pi , pi pi pipi . , pipi pi pi pi . pi pipi pi - pi pi .
1.1 -
1.1.1. X A P(X) pi X. A :(i) X A,(ii) A pi, A A,
Ac X \A A (iii) A pipi , A1, A2, ..., An A
nj=1Aj A.
1.1.2. () A pi X. A pipi , :
(iv) A,B A A \B A.(v) A1, A2, ..., An A
nj=1Aj A.
pi. (iv) pi Bc A pi (ii)
A \B = A Bc. (1.1)
pi pi pi (iii). (v) pi pi Acj A j pipi
nj=1
Aj =
nj=1
Acj
c (1.2)pi De Morgan.
6 -
() (i) 1.1.1 pi pi A A 6= .() (iii) 1.1.1 pi pi (v).
pi pi pi . pi pi . , :
1.1.3. X A P(X) pi X. A - :(i) X A,(ii) A pi, A A,
Ac X \A A (iii) A , An A, n = 1, 2, ...
j=1Aj A.
1.1.4. () - .
pi. A1, A2, ..., An A, Aj = X A, j n+ 1 nj=1
Aj =
j=1
Aj A.
() () () 1.1.2 A pi X - A 6= A pi .
1.1.5. () X . A1 = {, X} A2 = P(X) - X. A - X
A1 A A2. (1.3)
() X = N
A = {A N : A Ac pipi} (1.4)
A N, -.pi. A 6= A pi (-pi ). A1, A2, ..., An A, pipi:
Aj pi, Acj pipi,
nj=1
Acj =
nj=1
Aj
c
pipi. pinj=1Aj A.
1.1. - 7
pi Aj0 pipi, nj=1Aj A,
nj=1Aj Aj0 .
, pi A . -, An = {2n}, n = 1, 2, ... An A ( pipi),
j=1Aj / A.
() pi X = R A pi pipi pipi R -.
pi. , A 6= . pipi, I1 I2 R I1 I2 . , A =
ni=1 Ii B =
nj=1 Jj
A
A B =ni=1
mj=1
(Ii Jj) A.
pi, pi pi, A pipi . I R pi Ic R A. A = nj=1 Ij A
Ac =
nj=1
Icj A,
pi pipi. A . -, n In = (2n, 2n+ 1) A
n=1 In A
(;).
{An} pi X. {An} An An+1 n An An+1 n. pi pipi -:
1.1.6. X A pi X. A - X ( ) pi pi pi:(i) {An} A
n=1An A.
(ii) {An} A n=1An A.
(iii) {An} A n=1An A.
pi. A , . (Bn) A. (i). An =
nj=1Bj . A , An A
pipi An An+1 n. n=1An A
pi pi. pi
n=1
Bn =
n=1
An =n=1
Bn A. (1.5)
8 -
(ii) An =nj=1Bj . A , An A
pi An An+1 n. n=1An A.
pi
n=1
Bn =
n=1
An =n=1
Bn A. (1.6)
, pi (iii). pipi,
An = Bn \n1j=1
Bj (1.7)
pi An A n An . , pi pi
n=1An A. pi n=1
Bn =
n=1
An =n=1
Bn A. (1.8)
1.1.7. F P(X) pi X, pi - A X pi pi F , A - F A A A.pi. , pi pi (Ai)iI pi - X,
iI Ai - X ().
-
C = {A : - F A}. (1.9)
C 6= ( P(X) C) pi pipi pi (pi X)
A =C =
{B : B C} (1.10)
- X. pi F A A .
1.1.8. () - A pi pi pi pipipi - pi pi pi F (F).
pi - pi pi Lebesgue .
1.1.9. (X, d) 1 T - pi X. - pi pi T Borel pi X. Borel pi X - B(X).
1 Borel pi - pi pi .
1.1. - 9
:
A X G X, pi Gn , n = 1, 2, ... A =
n=1Gn.
B X F X, pi Fn , n = 1, 2, ... B =
n=1 Fn.
G F Borel. , B(X) pi pi .
1.1.10. F pi R pi :
1 = {(, b] : b R},
2 = {(a, b] : a < b, a, b R},
3 = {(a, b) : a < b, a, b R}.
B(R) = (F) = (1) = (2) = (3). (1.11)
pi.
B(R) (F) (1) (2) (3) B(R)
, , . B(R) F B(R) (F).pipi, 1 , 1 F (F) (1). a, b R a < b,
(a, b] = (, b] \ (, a] (1), (1.12)
2 (1) pi (1) (2). pi, (a, b) 3,
(a, b) =
n=1
(a, b 1n
] (2) (1.13)
pi 3 (2) (2) (3). pi , R 2 , T pi R T (3) pi
(3) (T ) = B(R),
pi .
pi pi pi- pi pi :
2pi , .
10 -
1.1.11. Fk pi Rk pi :
1 =
kj=1
(, bj ] : bj R, j = 1, 2, ..., k ,
2 =
kj=1
(aj , bj ] : aj < bj , aj , bj R, j = 1, 2, ..., k ,
3 =
kj=1
(aj , bj) : aj < bj , aj , bj R, j = 1, 2, ..., k .
B(Rk) = (Fk) = (1) = (2) = (3). (1.14)
1.2 Dynkin
1.2.1. X D P(X) pi X. D Dynkin :(i) X D,(ii) A,B D A B, B \A D (iii) D , (An)
D, n=1An D. 1.2.2. () - Dynkin.
() pi (i) 1.1.6 pipi D Dynkin pipi ( ) D -.() () . X A,B , pi X pi
A \B 6= , B \A 6= , A B 6= .
D = {, X,A,B,Ac, Bc}
Dynkin X , A,B D A B / D.() 1.1.7, pi Dynkin Dynkin , P(X) pi Dynkin D pi pi . 1.2.3. () Dynkin D pi pi pi pi-pi pi Dynkin pi pi pi ().
1.2. Dynkin 11
, P(X) () (), (1.15)
() -, Dynkin. pi .
1.2.4. pi X pipi- .
() = (). (1.16)
pi. , pi () -, () () pi . pi () - , 2.1.2 (), () pipi . , pipi
A B (), A () B (). (1.17) pi
() = {A X : A B (), B ()}. (1.18) pipi
() (). pi :
1. () pipi2. () Dynkin.
pi pi pi pi pi., P ()
P = {A X : A B (), B P}. (1.19): P Dynkin.
Dynkin pi :
(i) B P , X B = B P () X P .(ii) A1, A2 P A2 A1. , B P ,
(A2 \A1) B = (A2 B) \ (A1 B) (), A2 B,A1 B () A1 B A2 B. A2 \A1 P .
(iii) (An) P . , B P , ( n=1
An
)B =
n=1
(An B) (),
(AnB) (). ,pi
n=1An P .
12 -
P = () pi 2. 1 , pi- pipi . Dynkin, pipi
() .
, A () B A B () pi
()pi 1. pi .
1.2.5. pi pipi , 1.1.11pi
B(Rk) = (Fk) = (1) = (2) = (3).
pi. pi Fk, 1, 2{}, 3{} pipi .
1.3
.
1. X , A P(X) ( -) X C X.
AC = {A C : A A}
pi ( -) C.
2. X,Y , f : X Y B (. -) Y .
f1(B) = {f1(B) : B B}
pi (. -) X.
3. X C = {{x} : x X}.
pi (C).4. X (An) pi X.
lim supn
An = {x X : x pi pi An} (1.20)
lim inf
nAn = {x X : x An}. (1.21)
1.3. 13
()
lim supn
An =
n=1
k=n
Ak lim infn
An =
n=1
k=n
Ak. (1.22)
() (An) ,
lim supn
An = lim infn
An =
n=1
An
lim supn
An = lim infn
An =
n=1
An.
.
5. X F P(X) pi X. pi pi pi pi F . pipi F A(F).
6. I = {[a, b] : a, b R}.
(I) = B(R).7.
IQ = {(a, b) : a, b Q}. (IQ) = B(R).
8. X = {x1, x2, ...} . - X.
9. X,Y f : X Y .
A = {x X : f x} Borel pi X.
10. X fn : X R.
B = {x X : pi limn fn(x)}
Borel pi X.
11. X . R P(X) pipi . pipi R , -. pi :
() (. -) pipi (.) .
14 -
() (. -) R (. -) X R.() R -, {E X : E R Ec R} -.
() R -, {E X : E F R F R} -.
12. X F P(X). A (F) piCA F A (CA).(pi:
A = {A (F) : pi CA F A (CA)}
pi - F A. pi ;)13. X , - A X pi
pi C A = (C). pi B(R) pi. pipi, pi B(X), pi (X, d) .
.
14. X . M pi X X pi :
(i) , (An) M, n=1An M.
(ii) , (An) M, n=1An M. pi X, m() pi pi ( m() pi pi ). pi :
() Dynkin .
() pi X, m() ().() pi Dynkin.
() X,
m() = ().
15. X F pi X. A(F) (. 5) .
16. X A - X pi . :
() A pi pi .() A pi.
2
pi -, pi , . pi - , . pi pi :
1. 2. (Ai)iI pi , (pi pi pi
pi pipi I), .
- pi I 2 pi , Ai A i I, iI Ai A. pi :
R pi [a, b] b a. , . I 2 pipipi pi,
A =xA{x}
A R .
2.1
2.1.1. X A - X. : A [0,] :(i) () = 0 (ii) pi ( -pi), (An)nN -
A,
( n=1
An
)=
n=1
(An). (2.1)
16
pi , pi - pi ( -pi) .
pi, (X,A) , (X,A, ) (X,A) pi X. A A- . 2.1.2. X A - X. : A [0,] pipi pi :(i) () = 0 (ii) pipi pi, (Aj)nj=1 pipi -
A,
nj=1
Aj
= nj=1
(Aj). (2.2)
pipi pi .
2.1.3. (X,A) .() A A
(A) =
{n, A n pi , (2.3)
:
pi. () = 0 pi (ii), pi- An 6= , pi pi n = pipi pipi pipi -. pi pi .
.
() A A
(A) =
{0, A = , (2.4)
pi :
pi. () = 0 (ii), pi An = n, 0 = 0 pi n An 6= =.
() x X A A
x(A) =
{1, x A0, x / A (2.5)
x () Dirac x.
2.1. 17
, (X,A), + a , pi a R a 0, pi pi
(+ )(A) = (A) + (A), (a )(A) = a (A), A A. (2.6) pipi pi A A (A)
18
pi. pi pi 1.1.6.
Bn = An \n1j=1
Aj , n = 1, 2, ... (2.10)
Bn A, Bn , Bn An n=1
Bn =
n=1
An.
pi:
( n=1
An
)=
( n=1
Bn
)=
n=1
(Bn) n=1
(An),
pi .
2.1.7. (X,A, ) . :
(i) (An) A,
( n=1
An
)= limn(An). (2.11)
(ii) An A pipi (A1)
2.1. 19
, (Cn) A n=1
Cn = A1 \n=1
An.
pi (i), pi (n=1 Cn) = limn (Cn),
(A1 \
n=1
An
)= lim
n(A1 \An).
, pi 2.1.4 (ii)
(A1) ( n=1
An
)= (A1) lim
n(An)
(A1)
20
, :
( n=1
Bn
)=
( n=1
An
)= lim
n(An) =
limn
(nk=1
Bk
)= lim
n
nk=1
(Bk) =
n=1
(Bn),
(i) An pipi pi Bn. pi (ii).
An =
k=n
Bk (2.18)
pi An A n, (An) pipin=1
An =
(pi Bk , x X pi pi pi ). n ,
n=1
Bn = B1 B2 ... Bn1 An
pi pipi pi pi
( n=1
Bn
)=
n1k=1
(Bk) + (An).
n pi, pi pi
( n=1
Bn
)=
k=1
(Bk),
pi (ii) limn (An) = 0.
, pi pi pi.
2.1.9. (X,A, ) . :(i) pipi (X)
2.2. 21
2.1.10. () pipi, (A) < A A, pi .() -pipi, C A pi
C = X C =( n=1
An
) C =
n=1
(An C), (2.19)
(An C) (An)
22
D = A. pi :
D. pi A = (), D Dynkin,
A = () D . Dynkin - :
() X D pi pi (i).() A,B D B A
(A \B) = (A) (B) = (A) (B) = (A \B).( pi pipi.) , A \B D.
() (An) D. ,
( n=1
An
)= lim
n(An) = lim
n(An) =
( n=1
An
),
2.1.7. n=1An D.
pi D Dynkin pi .(ii) n = 1, 2, ... n, n : A [0,]
n(A) = (A Dn), n(A) = (A Dn), A A, (2.21) pi Dn . D ,
n(D) = (D Dn) = (D Dn) = n(D) pipi D Dn . pi
n(X) = (X Dn) = (Dn) = (Dn) = (X Dn) = n(X)
2.3. 23
pi. = {(, b] : b R}. pipi () = B(R) ( 1.1.10). pi,
(R) = limn ((, n]) = lim
n ((, n]) = (R)
24
(ii) pi (X,A) pi A , |A = .
(iii) A |A = .(iv) pi A = A ( = ).pi. , A A, pi E = F = A (i) 2.2.4,pi A A (A) = (A). pi, pi A A |A = . pipi :
(i) A 6= . A A E,F A E A F (2.24)
(F \ E) = 0. pipi Ec, F c A :F c Ac Ec. (2.25)
Ec \ F c = Ec (F c)c = F Ec = F \ E (2.26)
(Ec \ F c) = 0 Ac A, A pi-.
, (An) A (En), (Fn) A
En An Fn (2.27) (Fn \ En) = 0 n = 1, 2, .... ,
n=1
En n=1
An n=1
Fn, (2.28)
n=1En,
n=1 Fn A. pi, pi (
n=1
Fn
)\( n=1
En
)n=1
(Fn \ En) (2.29)
(( n=1
Fn
)\( n=1
En
))
( n=1
(Fn \ En))n=1
(Fn \ En) = 0.
pi pi n=1An A pi A pi -.
(ii) () = 0. (An) A, En , pi (i)pipi,
( n=1
An
)=
( n=1
En
)(2.30)
En (;) :
( n=1
En
)=
n=1
(En) =
n=1
(An).
2.4. 25
pi . pipi pi:
A - , pi B A
A B (B) = 0.
pi F A B F (F ) = 0. E = ,
E A F (F \ E) = (F ) = 0.pi A A.(iii) A |A = . , A A E,F A E A F :
(E) = (E) (A) (F ) = (F ).
(E) = (F ) pi (A) = (E), (A) = (A). pi = .
(iv) () A = A pi (ii) = pi. pi.() pi A A. A A. pi
E,F A E A F (F \ E) = 0. A \ E F \ E - pi pi A \ E A.
A = E (A \ E) A. (2.31)
2.3.4. () pipi (X,A) pi, - A.() (X,A) x X {x} A 6= P(X). pipi, Dirac = x pi.
pi. pi pi A = P(X). , A Xpi
A = (A {x}) (A {x}c) A {x} = {x} A A {x}c - pi {x}c pi ({x}c) = 0. , A 6= P(X) = A (iv) pi.
2.4
.
1. (X,A, ) . C : A [0,]
C(A) = (A C), A A (X,A).
26
2. (X,A, ) (An) A.
(lim infn
An) lim infn
(An) (2.32)
pipi (n=1An)
2.4. 27
7. (X,A, ) pi . pi A A B X A4B A (A4B) = 0, B A (A) = (B).
.
8. pi -pipi (R,P(R)) ((a, b)) = a < b R.
9. (X,A) {n} (X,A), n N A A n(A) n+1(A). A A
(A) = limnn(A).
(X,A).10. (X,A, ) -pipi (Ai)iI
A. A A JA = {i I : (A Ai) > 0} .
11. F X pipi (X,(F)). A (F) > 0 pi F F
(A4F ) < ,
pi A4F = (A \ F ) (F \A).12. (X,A, ) pipi (An) pi-
X pi pi > 0 (An) n N.() (lim supnAn) > 0.
() pi {kn} n=1
Akn 6= .
13. (X,A, ) . pipi A A (A) = pi B A B A 0 < (B) < . (X,A, ) pipi A A (A) = , M > 0 pi B A B A M < (B)
28
14. (X,A, ) .() A,B A A B (E4F ) = 0. A.() A,B A (A,B) = (A4B). A/ .
15. (X,A, ) . E X pi - E A A A A (A) 0 pi A A 0 < (A) < .() pi (An)nN A (An) > 0 n N.
17. (X,A, ) pipi E A. pi pi F E :
(i) A F , (A) > 0.(ii) F .(iii) F =
F , X \ F pi E -.
3
pi pi - . , pi pi . pi pi . pi :
1. : P(X) [0,] pi pi pi pi ( ). .
2. - A P(X) pi (X,A).
Le-besgue Rk pi Riemann.
, pi pi pi . :
1. pi A0 P(X).
2. pi - A pi pi A0.
. pi - .
3.1 Lebesgue
3.1.1. X . : P(X) [0,] :
(i) () = 0,(ii) , A B X (A) (B)
30
(iii) pipi ( -pipi), (An)nN pi X,
( n=1
An
)n=1
(An). (3.1)
pi pi .
3.1.2. () 1 : P(X) [0,]
1(A) =
{0, A = 1,
(3.2)
:
pi. (i) (ii) pi pi. (iii) , An = n pi 0=0, piAn0 , 1 (
n=1An) = 1
n=1 1(An) 1(An0) = 1, pi
.
pipi, pi |X| 2, 1 .() 2 : P(X) [0,]
2(A) =
{0, A 1,
(3.3)
:
pi. (i) pi . (ii), A B B pi, 2(A) 2(B) = 1, 2 0 1. B pi, A , 2(A) = 2(B) = 0. (iii), pi An0 pi, pi pi (ii). pi An ,
n=1An
2
( n=1
An
)= 0 =
n=1
2(An).
pi 2 .
: 2 ;1
1 pi .
3.1. Lebesgue 31
3.1.1 Lebesgue
Lebesgue R. , I = (a, b)
(I) = b a. (3.4)
A R , pi pi A pi pi , (In)n, In = (an, bn) A n=1 In (;). , n=1(bn an) pi pi A pi
(A) n=1]
(bn an), pipi (In)n A. (3.5)
pi :
3.1.3. Lebesgue : P(R) [0,] :
(A) = inf
{ n=1
(bn an) : an, bn R, A n=1
(an, bn)
}, (3.6)
A R. Lebesgue pi .
3.1.4. (i) : P(X) [0,] pi R.
(ii) a, b R a b
([a, b]) = ([a, b)) = ((a, b]) = ((a, b)) = b a. (3.7)(iii) I R, (I) =,pi. (i) :
() > 0, (, ). , (a1 = , b1 = an = bn n 2)
() 2.
> 0 , () = 0.() A B R, pi pi B
pi A. , (A) pi infimum pi . , B n=1(an, bn), A n=1(an, bn). {
((an, bn))n : B n=1
(an, bn)
}{
((an, bn))n : A n=1
(an, bn)
}, (3.8)
32
inf
{ n=1
(bn an) : B n=1
(an, bn)
} inf
{ n=1
(bn an) : A n=1
(an, bn)
}.
, (A) (B), .() (An) pi R.
( n=1
An
)n=1
(An).
n (An) = , pi. pi pi
n (An) 0 pi .
, pi .(ii) a, b R a b. ([a, b]) = b a. > 0. [a, b] (a 2 , b+ 2), ([a, b]) b a+ . > 0,
([a, b]) b a. ,
In = (an, bn), n = 1, 2, ... [a, b] n=1(an, bn) pipi
n=1
(bn an) b a.
[a, b] pi , (In)n pipi- pi, pi m N [a, b] mn=1(an, bn).
3.1. Lebesgue 33
. mn=1(bn an) b a.
pi pi m. m = 1 pi. pi m = k pi
[a, b] k+1n=1
(an, bn).
pi 1 i k + 1 a (ai, bi). pi i = 1, a1 < a < b1. b1 b, (a, b) (a1, b1]
b a < b1 a1 k+1n=1
(bn an).
pi b1 < b
[b1, b] k+1n=2
(an, bn)
, pi pi pi
b b1 k+1n=2
(bn an).
,
b a < b a1 = (b1 a1) + (b b1) k+1n=1
(bn an),
pi . pi .
pi n=1(bn an)
mn=1(bn an) b a,
([a, b]) = b a. (a, b) , a < b a + 1n b 1n
n. ,[a 1n , b+ 1n
] (a, b) [a, b] pi
([a 1
n, b+
1
n
])= b a 2
n ((a, b)) b a.
n pi ((a, b)) = b a. pi , .
(iii) I pi pi , n pi an R (an, an + n) I. (I) n n pi .
pi pi, pi Lebesgue Rk. Rk
I =
kj=1
(aj , bj) = (a1, b1) (a2, b2) ... (ak, bk) (3.9)
34
pi aj < bj R. I pi
v(I) = (b1 a1)(b2 a2)...(bk ak). (3.10)
, Rk I =kj=1 Ij , pi
I1, I2, ..., Ik R Ij (pi 0 = 0). pi 3.1.3 pi :
3.1.5. Lebesgue k : P(Rk) [0,] Rk :
k(A) = inf
{ n=1
v(In) : In Rk A n=1
In
},
(3.11) A Rk.
pi 1 = . , pi,
k = .
pi k ( 3.1.4) :
3.1.6. (i) pi Rk pi pipi Rk.
(ii) Ij , j = 1, 2, ..., n Rk, I =nj=1 Ij
J Rk I J . nj=1
v(Ij) v(J) (3.12)
pipi I
nj=1
v(Ij) = v(I). (3.13)
I1
I2
I3J
I1
I2 I3
I4
I5
I
3.1: 3.1.6 (ii)
3.1. Lebesgue 35
pi. (i) 6= . A,B . , A =ni=1 Ii B =
mj=1 Jj , pi (Ii)i (Jj)j
Rk.
A B =i,j
Ii Jj
pi (IiJj)(i,j) pi Rk. AB , pipi .
pi , pi I Rk Ic Rk \ I .
I
J1 J2 J3
J4
J5J6J7
J8
3.2: I Ic .
, Ac =ni=1 I
ci pi pipi. pi .
3.1.7. pipi - Rk ( ). = , pi pi pi . , pipi Rk pi pipi .
(ii) pi pi n. n = 1 pi.pi n = m pi n = m + 1. Rk , I pi m , m+ 1, pi pi.
pi pi Ij 6= j, pipi n = m. j = 1, 2, ...,m+ 1
Ij =
k=1
Ij,,
pi Ij, R.
I1 Im+1 =(
k=1
I1,
)(
k=1
Im+1,
)=
k=1
(I1, Im+1,) = .
, pi 1 0 k I1,0 Im+1,0 = . pi pi I1,0 Im+1,0 . I1,0 ,
36
J1 = {(x1, x2, ..., xk) Rk : x0 < } J1 = {(x1, x2, ..., xk) Rk : x0 > }.
I1 Im+1
J1 J2
3.3: I1 Im+1 pi pipipi x0 = .
, J1 Im+1 =
J1 I =mj=1
(J1 Ij) J1 J
, J2 I1 =
J2 I =m+1j=2
(J2 Ij) J2 J.
pi pi pi,
mj=1
v(J1 Ij) v(J1 J) m+1j=2
v(J2 Ij) v(J2 J)
v(J) = v(J1 J) + v(J2 J) mj=1
v(J1 Ij) +m+1j=2
v(J2 Ij) =
m+1j=1
v(J1 Ij) +m+1j=1
v(J2 Ij) =m+1j=1
(v(J1 Ij) + v(J2 Ij) =m+1j=1
v(Ij),
pi pi v() = 0 v(K) = v(K J1) + v(K J2) (3.14)
K Rk. I , J1 I J2 I. ,pi pi pi pi,
mj=1
v(J1 I) = v(J1 I) m+1j=2
v(J2 Ij) = v(J2 I). (3.15)
pi pipi pi v(I) =m+1j=1 v(Ij), .
3.1. Lebesgue 37
pi () , - 3.1.4:
3.1.8. (i) : P(Rk) [0,] pi Rk.
(ii) I Rk (I) = v(I).
pi. pi pi pi 3.1.4 pi :
: K =k=1[a, b] pi Rk I1, I2, ..., In
K nj=1 Ij , v(K)
nj=1
v(Ij). (3.16)
pi , pi 3.1.7 (i) pi
nj=1 Ij pi (ii)
pi . ,
Ej = Ij \j1i=1
Ii, j = 1, 2, ..., n (3.17)
pi (pi pi ),Ej Ij j pipi
nj=1Ej =
nj=1 Ij . , pi
3.1.7 (i), Ej pipi Rk. Jt, t = 1, 2, ...,m . , Jt (;)
K nj=1
Ij =
nj=1
Ej =
mt=1
Jt.
, K =mt=1(K Jt), pi K Jt . ,
pi (ii) pi
v(K) =
mt=1
v(K Jt) mt=1
v(Jt) =
nj=1
{t:JtIj}
v(Jt) =
nj=1
v(Ej) nj=1
v(Ij),
pi .
3.1.9. A Rk (A) = 0.pi. A = {xn : n = 1, 2, ...}. , > 0
A n=1
k=1
(xn()
2n, xn() +
2n
)(3.18)
, pi
(A) n=1
k=1
((xn() +
2n) (xn()
2n))
=
n=1
(2)k
2nk=
k
1 1/2k .
> 0 pi (A) = 0.
38
3.1.2
Lebesgue - ( - pipi) pi pi pi pi . pi . pi :
3.1.10. X 6= . C P(X) pi X - X
(i) C (ii) pi X1, X2, ... C X =
n=1Xn.
3.1.11 ( ). X 6= , C - X : C [0,] () = 0. : P(X) [0,]
(A) = inf
{ n=1
(Cn) : Cn C A n=1
Cn
}(3.19)
A X X.
pi , pi C - . pi pi 3.1.4 (i) pipi . pi pi.
pi. 3.1.1 :
(i) n () n () = 0. () = 0.(ii) A B X , B pi C A, {
(Cn)n : Cn C B n
Cn
}{
(Cn)n : Cn C A n
Cn
}.
pi, pi (A) (B) (;).(iii) pipi . (An)n pi X. (
nAn)
n (An).
n (An) = pi. pi pi
n (An) (I).14. pi Lebesgue A R (A) > 0
(A I) < (I) I.15. X A X. A
A A A. 0 premeasure A . :() A X > 0, pi B A A B (B) (A) + .() (A)
3.5. 49
16. X pi (X,M). E,G X, G E :
E G, G M A M A G \ E (A) = 0.
() G1 G2 E X, (G14G2) = 0.() E G, G M (E) = (G), G E.
17. (An) pi [0, 1]
lim supn
(An) = 1.
(0, 1) pi pi (Akn) (An)
( n=1
Akn
)> .
.
18. {qn} Q [0, 1]. > 0
A() =
n=1
(qn
2n, qn +
2n
).
, A =j=1A(1/j).
() (A()) 2.() < 12 [0, 1] \A() .() A [0, 1] (A) = 0.() Q [0, 1] A A pi.
19. A Lebesgue pi Rk (A) 0 n N.() (lim supnAn) > 0.
() pi {kn}
n=1
Akn 6= .
20. {qn} . x R (pi Lebesgue) :
pi k = k(x) N n k |x qn| 1/n2.
50
21. A R pi pi a > 0, {x A : |x| > a} pi.
(A) =
0, A
1, A pi pi pi
, A pi pi pi.
R
M = {A R : A Ac }.
A R ;22. Lebesgue A R (A) > 0.
AA = {x y : x, y A} pi 0.
4
Lebesgue
pi pi pi Lebesgue Rk. Lebesgue .
B(Rk) M P(Rk)
pi pi . pi pi- , pi Rk pi Lebesgue Lebesgue pi Borel.
4.1 Lebesgue
4.1.1. (X, d) , A - X A B(X) (X,A). :
(i) (K)
52 Lebesgue
4.1.2. Lebesgue Rk . pipi,
(A) = sup{(K) : K pi K A}, A M (4.3)
( ).
pi. 3.2.6 B(Rk) M pi (i)-(iii) 4.1.1:
(i) K Rk pi . , pi J Rk K J . , pi
(K) (J) = v(J)
4.1. Lebesgue 53
pipi pi , > 0 L pi L A U L \A U pipi
(U \ (L \A)) < .
, K = A \U . K A K = L \U (;),pi K . pipi, A \K U \ (L \A) pi
(A \K) (U \ (L \A)) < .
pipi ( A pi ),
An = A B(0, n), pi B(0, n) = {x Rk : x2 < n}, n = 1, 2, ... (4.4)
, (An) An Rk.:
(A) = supnN
(An) = supn
sup{(K) : K An pi}
sup{(K) : K A pi}.
pi Lebesgue (Rk,M , ) (Rk,B(Rk), ). pi 3.2.6 B(Rk) M , pi- pi:
4.1.3. Lebesgue (Rk,M) pi Lebesgue (Rk,B(Rk)).
pi. pi -B(Rk) M = B(Rk) :
A M pi E,F B(Rk), E A F (F \ E) = 0. (4.5)
() A M . pi (A) < . , pi pipipi (Kn), (Gn) Kn pi Gn , Kn A Gn
(Gn) (Kn) = (Gn \Kn) < 1n.
(pi ) E =nKn F =
nGn. , E A F ,
E,F B(Rk)
(F \ E) (Gn \Kn) < 1n, n = 1, 2, ...
(F \ E) = 0, A B(Rk). A M
A =
n=1
An, pi An = A B(0, n).
54 Lebesgue
, An pi pipi An B(Rk). B(Rk) -, A B(Rk) () pi-.
() A Rk pi E,F B(Rk) E A F (F \ E) = 0, A = E (A \ E). , (A \ E) = 0 pi A \ E M pipi E B(Rk) M . A M .
4.1.4. pi -:
A M pi E A G (E \A) = 0 (4.6)
A M pi F A F (A \ F ) = 0, (4.7) pi pi .
4.1.5. (X, d) . (X,B(X)) Borel X. 4.1.6. Lebesgue Borel Rk
(I) = v(I), I Rk. (4.8)
pi. ( 2.2.1).
= {I Rk : I }. pipi () = B(Rk) . Borel Rk (I) = v(I) I , (I) = (I) I . pipi Rk = n=1[n, n]k, pi ([n, n]k)n ([n, n]k) = ([n, n]k) = (2n)k < n. ,pi 2.2.1 = .
4.1.7. pi
Borel Rk (P)
pi
(Rk,M) (P)pi pi ( 2.3.3 (iii)). pi, pi Lebesgue Borelpi pi (Rk,M).
pi pi 3 Littlewood2. pi
2 Egorov Luzin pi 7 8 .
4.1. Lebesgue 55
Rk pipi .
, . pi :
4.1.8. A Lebesgue (A) < . > 0 pi J1, J2, ..., Jm
(A4(J1 J2 ... Jm)) < .3 (4.9)
pi. pi Lebesgue, (In)n A
n In
n=1
v(In) < (A) +
2.
v(In) , pi N N
n=N+1
v(In) 0
B(0, ) AA. (4.30)
pi. , pi pi (A) < . pipipi (A) = pi (pi pi) B M B A 0 < (B) < . , pi > 0 B(0, ) B B B(0, ) AA.
pi 0 < (A) < pi. pi Lebesgue. > 0. , 4.1.2, piK Rk pi G Rk K A G
(G) < (A) + , (K) > (A) . (4.31)
K G pi
:= dist(K,Gc) > 0. (4.32)
, z K B(z, ) G. pi .pi x Rk x < A. pi :
. B(0, ) K K. x Rk x < . z1, z2 K x = z1 z2. , z K x + z K. pi pi , K (K + x) = :
(K (K + x)) = (K) + (K + x) = 2(K) > 2(A) 2
pi Lebesgue . K G pipiK + x G, y K + x, pi z K y z = x < y G. pi,
(K (K + x)) (G) < (A) + ., :
2(A) 2 < (A) +
(A) < 3.
> 0 pi , (A) = 0 pi pi.
62 Lebesgue
4.3
pi - B(Rk) M pi-
B(Rk) M P(Rk). (, pipi R pi pi pi Borel) pi. pi pi- . pipi , pi pi.
pi: X = {Xa : a A} , pi . , pi E pi pi xa pi Xa. , pi pi- f : A f(a) Xa a A.. pi, , pi - pi (Zermelo-Fraenkel) .
4.3.1 (Vitali). pi pi R.
pi. R :x y x y Q. (4.33)
R Ex = {y R | y = x+ q pi q Q}. (4.34)
X = {Xa : a A} , pi pi E = {ya : a A} R pi pi ya pi Xa. , a 6= b A ya yb / Q. {qn : n N} Q
En := E + qn, n N. (4.35) En pi :
1. n 6= m En Em = . , pi ya, yb E ya + qn = yb + qm, 0 6= ya yb = qm qn Q, pi pi pi pi E.
2. R =n=1En. , x R pi a A x Xa.
x = ya + q pi q Q. , pi n = n(x) N q = qn, , x = ya + qn En.
pi E . , En = E + qn n N (En) = (E). pi En pi pi , pi
+ = (R) =n=1
(En) =
n=1
(E).
4.3. 63
pi, (E) > 0. pi Steinhaus, EE pi (, ) pi > 0. pi, E E pi pi pi 0: x 6= y E xy , pi pi E. pi E .
4.3.2. pi pi pi:
A R , pi E A. pi .
pi, pi pi, pi pi E [0, 1], pi Steinhaus.
pi. [0, 1] :
x y x y Q. (4.36)
, , x y [1, 1]. [0, 1]
Ex = {y [0, 1] | y = x+ q pi q [1, 1] Q}. (4.37)
X = {Xa : a A} , pi pi E = {ya : a A} [0, 1] pi pi ya pi Xa. , a 6= b A ya yb / Q. {qn : n N} Q [1, 1]
En := E + qn, n N. (4.38) En pi :
1. En [1, 2].2. n 6= m En Em = .3. [0, 1] n=1En. , x [0, 1] pi a A x Xa.
x = ya + q pi q Q [1, 1]. , pin = n(x) N q = qn, , x = ya + qn En.
pi E . , En = E + qn n N (En) = (E). pi En pi pi , pi
1 = ([0, 1]) ( n=1
En
)=
n=1
(En) =
n=1
(E) 3,
pi pi 0 ( (E) = 0) + ( (E) > 0). pi, E .
64 Lebesgue
4.4 pi Borel
pi pi pi pi-pi. pi , pi pi R pi Borel. pi , pi pi pi Cantor-Lebesgue pi pi 4.4.2. pi Cantor pi .
4.4.1 Cantor
1. Cantor C0 = [0, 1] . -
(1/3, 2/3
). C1 pi pi,
C1 =
[0, 1/3
] [2/3, 1]. C1 pi . pi
[0, 1/3
]
[2/3, 1
] , pi pi ,
. C2 pi pi,
C2 =[0, 1/9
] [2/9, 1/3] [2/3, 7/9] [8/9, 1]. pi, n = 1, 2, . . . Cn (Cn) :
1. C1 C2 C3 .2. Cn 2n , pi pi
1/3n.
Cantor
C =
n=1
Cn. (4.39)
. [k/3n, (k+1)/3n
], n N, k = 0, 1, . . . , 3n
1, .
2. Cantor C , pi pi pi Cn (pi pi pi pi ).pi C , .pipi, C :
(1) C , C C.
pi. C . x C C pi x C pi
4.4. pi Borel 65
In(x), n = 1, 2, . . ., x In(x), In(x) Cn `(In(x)) =
13n . (n(x)) (n(x))
In(x) pi C, pi x, pi . , x C.
(2) C 0.
pi. n N C Cn (Cn) = 2n3n , Cn 2n , pi pi 13n . ,
(C) (Cn) = 2n
3n
n N, pi (C) = 0.
. , C pi .
(3) C pi.
pi. pi pi, pi R pi. C , pi . pi, pi pi C pi pi .
pi pi pi pi C
{0, 2}N = {(n)n=1 | n, n = 0 n = 2}. (4.40) {0, 2}N pi ( pi Cantor). , C pi. pi :
x C pi In(x), n =1, 2, . . ., : I1(x) I2(x) , n, x In(x) In(x) pi 13n pi pi Cn.
(xn)n=1
{0, 2}N :() n = 1: x1 = 0 I1(x) =
[0, 1/3
](, x [0, 1/3]) x1 = 2
I1(x) =[2/3, 1
](, x [2/3, 1]).
() pi : n, In(x) =[k/3n, (k+1)/3n
] In+1(x)
pi [k/3n, (k/3n)+(1/3n+1)
], [(k/3n)+(2/3n+1), (k+1)/3n
]:
pi pi x. xn+1 = 0 In+1(x) pi , xn+1 = 2 In+1(x) .
x 6= y, pi n In(x) 6= In(y), pipi |x y| 13n n N. n0 pi pi In0(x) 6= In0(y), pi xn pi xn0 6= yn0 , (xn)
n=1 (
yn)n=1 . pi
pi : C {0, 2}N (x) = (xn)n=1 pi ., (n)n=1 pi 0 2,
(In)n=1 I1 I2 , n In pi 13n pi pi Cn:
() n = 1: I1 =[0, 1/3
] 1 = 0 I1 =
[2/3, 1
] 1 = 2.
66 Lebesgue
() , In+1 pi pi 1
3n+1 In pi pi Cn+1: n+1 = 0, n+1 = 2.
In 0, :
{x} =n=1
In.
( -). In Cn n, x C. pi, In(x) = In n, pi pi In
(n)n=1 = (
xn)n=1 = (x).
pi pi {0, 2}N, C pi.
pi pi Cantor.
3. pi (an)n=1 an {0, 1, 2} n N, n=1
an3n x [0, 1]. x =
n=1
an3n an {0, 1, 2}
n, n=1
an3n ( (an)
n=1) pi
x. x = (a1, a2, . . .) x =n=1
an3n .
x [0, 1] pi. (an)
n=1 pi pi : [0, 1] pi
[0, 1/3], (1/3, 2/3) [2/3, 1].
a1 =
0 , x [0, 1/3]1 , x (1/3, 2/3)2 , x [2/3, 1].
, pipi
a13 x a1
3+
1
3. (4.41)
pi x [0, 1/3]. pi[0, 1/9], (1/9, 2/9), [2/9, 1/3] a2 = 0, 1 2 x , pi . a2 x (1/3, 2/3) x [2/3, 1], pipi
a13
+a232 x a1
3+a232
+1
32. (4.42)
pi an pi n
nk=1
ak3k x
nk=1
ak3k
+1
3n. (4.43)
pi
0 xnk=1
ak3k 1
3n,
4.4. pi Borel 67
pi k=1
ak3k
x,
x =
k=1
ak3k.
x 6= y pi x pi y, pi .
pi x [0, 1] pi pi-. pi, x = 1/3
1
3=
1
3+
k=2
0
3k
1
3=
k=2
2
3k.
( pi pi (an)n=1 pi pi pipi, pi).
, : x [0, 1] pi- x : x = k/3n pin N pi 1 k 3n ( ).
pi pi pi Cantor.
4.4.1. x [0, 1]. , x C x pi pi pi 0 2. 2
pi. x [0, 1]. (an) pi pi pi pi- pipi, : x C an 6= 1 n. pi x C x pi pi pi 0 2. pi .
4. pi pi Borel pi -:
4.4.2. X , |B(X)| c,pi c = |R| . , X = R, |B(X)| = c.
pi. X pi pi D ={x1, x2, ...} X. ,
C = {B(xn, qm) : n,m = 1, 2, ...},
pi {qm : m = 1, 2, ...} Q(0,) pi X, X C - . pi, (C) B(X) = (C). piB B(X) : B0 = C,
B =
68 Lebesgue
B+1 pi B. pi B pi pi
B(X) ={B : }.
pi, pi, B(X) pi pi , |B(X)| c.
pipi R , pi : R B(R) (x) = (, x) 1 1 c = |R| |B(R)|. |B(R)| = c pi pi Schroder-Bernstein.
4.4.3. pi Lebesgue pi R pi Bo-rel.
pi. C Cantor, pi pi (C) = 0 pi
P(C) M .pi, |M | |P(C)| > |C| = c. pipi pi : |B(R)| = c.
4.4.2 Cantor-Lebesgue
Cn pi pi C Cantor. n N fn : [0, 1] [0, 1] . Jn1 , . . . , J
n2n1 pi [0, 1] \ Cn,
fn(0) = 0, fn(1) = 1, fn(x) = k2n x Jnk , pi
pi pi Cn pi .
13
23
1
12
1 f1
19
29
13
23
79
89
1
14
12
34
1 f2
127
227
19
29
727
827
13
231927
2027
792227
2327
892527
2627
1
14
12
34
1 f3
4.2: Cantor-Lebesgue
4.4. pi Borel 69
pi, C1 = [0, 1/3] [2/3, 1]. f1 1/2 (1/3, 2/3), [0, 1/3] f(0) = 0 f(1/3) = 1/2, [2/3, 1] f(2/3) = 1/2 f(1) = 1. , [0, 1]\C2 pi pi : (1/9, 2/9) f2 1/4, (1/3, 2/3) f2 1/2, (7/9, 8/9) f2 3/4, pi C2 pi , pi f2(0) = 0 f2(1) = 1.
4.4.4. {fn}n=1 - f : [0, 1] [0, 1]. f pi [0, 1]. C f (f(C)) = 1.
pi. pi {fn} :1. fn , fn(0) = 0 fn(1) = 1.
2. Jnk pi pi pi n- C, fn Jnk ,
fn fn+1 fn+2
Jnk .
3.
fn+1 fn 12n, n = 1, 2, 3, . . . .
pi {fn} C[0, 1]: m > n
fm fn m1k=n
fk+1 fk m1k=n
1
2k 1
2n1 0
m,n . C[0, 1] pi pi , pi f : [0, 1] R fn f .
, fn f [0, 1]. fn - fn(0) = 0 fn(1) = 1, pi f , f(0) = 0 f(1) = 1. , f pi [0, 1].
, f(C) = [0, 1]. , pi {fn} pi f J pi C, pi J pi C. f pi [0, 1], y [0, 1] f(x) pi x C. pi f(C) = [0, 1] (f(C)) = 1.
. ([0, 1] \ C) = 1 f (x) = 0 x / C., x / C x pi J pi f . pi, f pi x f (x) = 0. , f pi , pi pi f pi [0, 1] pi [0, 1].
pi CantorLebesgue, pi pi pi pi Borel. :
70 Lebesgue
4.4.5. A Borel R f : A R -. , Borel B R, f1(B) = {x A : f(x) B} Borel.
pi.
A = {B R : f1(B) Borel}. (4.44) B pi R, f1(B) A, f . A Borel, pi f1(B) Borel ( ).
A - pi -. A - pi , pi Borel - B(R) pi A. pi A pi f1(B) Borel B R Borel. 4.4.6. pi Lebesgue pi Cantor, pi Borel.
pi. g : [0, 1] [0, 2] g(x) = f(x) + x, pi f CantorLebesgue. g , pi ( g1).
g(C) (g(C)) = 1. , g(C) pi C, . pi, g pi J [0, 1] \ C {f(J)} + J , . (g([0, 1] \ C)) = (J) = 1. pi (g(C)) = 1.
g(C) , pi pi M g(C)., K = g1(M) Lebesgue pi C pi . , K Borel: , pi 4.4.5 M = (g1)1(K) Borel Borel . pi, M Lebesgue .
4.5
.
1. pi A R (A) > 0 pi.
2. pi Lebesgue pi A R2 pi1(A) Lebesgue , pi pi1(x, y) = x (x, y) R2 pi pi .
3. C Cantor, 14 C, pi pi 14 pi pi Cantor.
4. A R, a R > 0. pi t (, ) a+ t A a t A. (A) .
.
5. E, F pi pi Rk E F (E) < (F ). a ((E), (F )) pi pi K E K F (K) = a.
4.5. 71
6. A = Q [0, 1]. :() > 0 pi {In}n=1
A n=1
In n=1
(In) < .
() pipi {In}mn=1
A mn=1
In mn=1
(In) 1.
7. {qn}n1 . pi B (B) = 0 x R \ B : pi k =k(x) N n k |x qn| 1/n2.
8. () f : R R , pi Lipschitz [a, b] R.
(i) f pi Lebesgue Lebesgue.
(ii) f pi Lebesgue Lebesgue - .
() f : R R pi Lebesgue Lebesgue ;
9. () G , pi Rn. pi - {Bj} G pi pi : G pi pi Bj
j=1 (Bj) 1
j=1((Bj))
p 0.
72 Lebesgue
.
13. A M x R
(A, x) = limt0+
(A (x t, x+ t))2t
,
pi. (A, x) pi A x.
() (Q, x) = 0 (R \Q, x) = 1 x R.() 0 < < 1. A R (A, 0) = .
14. pi {An} pi R
( n=1
An
)0. E + F pi .
18. E x R pi {sin(2nx)}n=1 . (E) = 0.
19. f : [0, 1] R f(0) = f(1). A = {t [0, 1] : pi x [0, 1] f(x+ t) = f(x)}.
() A , .
() B = {t [0, 1] : 1 t A}, A B = [0, 1].() (A) 1/2.
20. pi : (E) > 0 x, y E pi 12 (x+ y) E, E .
21. A R (A) > 0. (R \ (A+Q)) = 0.
22. Lebesgue E [0, 1] : J [0, 1]
(J E) > 0 (J \ E) > 0.
5
, pi pi , - pi . pi , , .... - . pi pi Lebesgue.
, pi , Lebesgue pi-
Xf f X pi
t1k=0
yk({x X : yk f(x) < yk+1}) (5.1)
pi {y0 < y1 < ... < yt} pi f . pi, pi :
1. f pipi (X,A) [,].
2. Bk = {x X : yk f(x) < yk+1}
pipi ( pipi pi yk, yk+1), - A.
, pi Lebesgue pi pi (X,A) (5.1), pi pi pi.
5.1
pi :
5.1.1. (X,A) .
74
(i) f : X [,] pi A ( A-)
[f b] := f1([, b]) = {x X : f(x) b} A, b R. (5.2)
(ii) (X,A) f - A-.(iii) , X = Rk - Lebesgue -
.
(iv) X f B(X)-, f Borel .
pi -:
5.1.2. (X,A) f : X [,] . :
1. f A-.2. [f < b] = {x X : f(x) < b} A b R.3. [f b] = {x X : f(x) b} A b R.4. [f > b] = {x X : f(x) > b} A b R.
pi. (i) (ii) b R,
[f < b] =
n=1
[f b 1
n
](5.3)
, x X f(x) < b pi n f(x) b 1n . [f < b] A.(ii) (iii) b R
[f b] = [f < b]c (5.4) [f b] A.(iii) (iv) pi pi pi, b R
[f > b] =
n=1
[f b+ 1
n
](5.5)
pi [f > b] A.(iv) (i) b R
[f b] = [f > b]c (5.6) [f b] A pi f A-.
5.1. 75
5.1.3. () (X,A) B X. B : X R
B(x) =
{1, x B0, x / B (5.7)
B. B A- B A.pi. pipi pi pi
[B b] =
, b < 0Bc, 0 b < 1X, b 1.
(5.8)
() f : Rk R
f f Borel f Lebesgue . (5.9)
pi. f , b R [f b] = f1((, b]) ( (, b] ) Borel. (- pipi pi 4.4.5 pi pi-.) pi pi pi B(Rk) M .
() I R f : I R f Borel .
pi. b R.
a = sup[f b] = sup{x I : f(x) b}.
f , t, s I f(t) b s < t f(s) b. pi
[f b] ={I (, a], a I f(a) bI (, a), (5.10)
pipi [f b] B(R). pi .
pi :
5.1.4. (X,A) C X.
AC = {A C : A A}. (5.11) pi AC - C () pi ( pi) A C.
76
f : C [,] AC-,
[f b] AC , b R. C A pi AC = {A A : A C}
f [f b] A, b R. :
5.1.5. (X,A) f :X [,].(i) f A- C X f |C AC-.(ii) (Cn) A X =
n=1 Cn,
f A f |Cn ACn n. (5.12)
pi. (i) b R [f |C b] = {x C : f(x) b} = C [f b] AC .
f |C pi .(ii) b R
[f b] =n=1
(Cn [f B]) =n=1
[f |Cn b
] A, f .
5.1.6. (X, d) Y X. :(i) B(X)Y = B(Y ) (ii) f : X [,] Borel f |Y : Y [,]
Borel .
pi. (i) pi B Y Borel Y piA Borel X B = A Y . Y G Y G X .
A = {A X : A Y B(Y )}. (5.13) () A - X pi pi X. B(X) A, pi () pipi. , pi U Y U = G Y piG X U B(X)Y . B(X)Y - pi pi Y pi Borel Y B(X)Y , .
(ii) (i) pi f |Y B(X)Y pi (i) pi Borel .
5.2. 77
- pi -.
5.1.7. (X,A) . f :X [,] :(i) f .
(ii) f1(G) A G R .(iii) f1(F ) A F R .(iv) f1(B) A B B(R).pi.
F = {A [,] : f1(A) A}. (5.14) pi F - [,]. pi . pi (i)-(iv) :
(i) F pi (, b], b R.(ii) F pi pi R.(iii) F pi pi R.(iv) F pi Borel pi R., pi B(R) - pi pi pi (, b], pipi F -,pi (iv) pi (i)-(iii) pi .
5.2
pi . pi pi - pi pi .
5.2.1. (X,A) f, g : X [,] . :
(i) [f < g] = {x X : f(x) < g(x)} A,(ii) [f g] = {x X : f(x) g(x)} A (iii) [f = g] = {x X : f(x) = g(x)} A.pi. (i) pi pi R pi : x X
f(x) < g(x) pi q Q f(x) < q < g(x)., pi
[f < g] =qQ
([f < q] [g > q]) (5.15)
78
pi A f, g Q .(ii)
[f g] = [g < f ]c pi pi (i).
(iii) [f = g] = [f g] \ [f < g]
pi pi (i) (ii).
5.2.2. (X,A) f, g : X [,] - .
(i) f g = max{f, g} f g = min{f, g} .(ii) f+ = f 0 f = (f) 0 .
pi. (i) b R [f g b] = [f b] [g b] A (5.16)
[f g b] = [f b] [g b] A. (5.17)
.
(ii) pi pi (i) 0 f .
f
f+
f
5.1: f+ f
5.2. 79
5.2.3. f+ f pi pi. f . pipi f :
f = f+ f |f | = f+ + f (5.18)
pi pi .
5.2.4. (X,A) fn : X [,] .
(i) supn fn infn fn .
(ii) lim supn fn lim infn fn .
(iii) {fn} f , f .
pi. (i) b R pi
[supnfn b] =
n=1
[fn b] A (5.19)
[infnfn < b] =
n=1
[fn < b] A (5.20)
pi.
(ii) (an)
lim supn
an = infnN
(supkn
ak
) lim inf
nan = sup
nN
(infkn
ak
)(5.21)
lim supn
fn = infnN
(supkn
fk
) lim inf
nfn = sup
nN
(infkn
fk
).
pi, pi (i) lim supn fn lim infn fn .
(iii) f = limn fn, f = lim supn fn = lim infn fn pi pi (ii).
pipi . ( ), . pi Riemann pi pi Lebesgue.
80
5.2.5. (X, d) . f : X R Baire-1 pi fn : X R f fn.
f : X R Baire-2 Baire-1 n 2 Baire-n Baire-(n-1) .
pi pi, pi Borel - (iii) , Baire-n Borel-.
5.2.6. f : R R pi , f : R R Borel .
pi. x R
f (x) = limh0
f(x+ h) f(x)h
= limnn
(f(x+
1
n) f(x))
pi . f Baire-1 Borel.
pi pi pi :
5.2.7. (X,A) , f, g : X [0,] a 0. :(i) a f .(ii) f + g
pi. (i) a = 0 a f pi pi. a > 0 , b R
[af b] =[f b
a
] A
a f .(ii) b R
[f + g < b] =qQ
([f < q] [g < b q])
pi pi (i) 5.2.1. , pi f + g .
5.2.8. (X,A) , f, g : X R - a R. (i) a f .(ii) f + g f g .
5.3. pi 81
(iii) f2 f g .(iv) g(x) 6= 0 x X, fg .(v) |f | .
pi. (i) pi (i) pi pi pipi a < 0. b R
[a f b] =[f b
a
] A
.
(ii) pi pi (ii) pi , g .
(iii) pi pi f2. b 0
[f2 < b] =
b > 0 [f2 < b] = [f
b] A. (5.22)
, pi f2 . f g ,
f g = (f + g)2 (f g)2
4(5.23)
pi pipi .
(iv) (iii) 1/g . A = [g > 0] A b R [
1
g b]
= ([bg 1] A) ([bg 1] Ac) A, (5.24)
b g . 1/g .(v)
|f | = f+ + f
( 5.2.3) pi |f | pi 5.2.2 (ii).
5.3 pi
pi pi Lebesgue pi Riemann. :
5.3.1. (X,A) . s : X R pi s(X) pipi.
82
pi
s =
nj=1
ajAj (5.25)
pi {A1, A2, ..., An} pi X aj R., pi.
, pi pi s, pi - . s(X) = {a1, a2, ..., an} ai 6= aj i 6= j Aj = {x X : s(x) = aj} {A1, A2, ..., An} X pi
s =
nj=1
ajAj .
pi s s .
pi pi ( 5.3.3) pi . pi :
5.3.2. (X,A) f : X [0,] .
(i) pipi P [0,) 0 P , pi
P = {0 = a0 < a1 < ... < an},
sP =
nj=0
ajAj , (5.26)
pi Aj = [aj f < aj+1] 0 j n 1 An = [f an]. sP pi 0 sP f x X f(x) < an
0 f(x) sP (x) < P, (5.27)pi P = max{aj+1 aj : 0 j n 1}.
(ii) P,Q [0,) pipi 0 P P Q sP sQ.pi. pi pi pi pipi sP . pi pi pi, f (sn) pi pi pi ( pi ). sP
pi :
1. 1 P = {0 = a0 < a1 < ... < an} [0,] X pi f . Aj .
2. pi Aj = [aj f < aj+1] sP pi pi pi f , aj . , pi pi pi f .
5.3. pi 83
a2a3a4a5a6a7
...
X
5.2: pi pi
pi :
(i) sP pi . , {A0, A1, ..., An} X pi pi Aj aj . x Aj , 0 j n
sP (x) = aj f(x),pi Aj pipi j < n ( f(x) < an)
f(x) sP (x) = f(x) aj < aj+1 aj Ppi P.(ii) sP pi (5.26). pipi pi Q = P {a} pi a > 0, a / P . pipi pi pi |Q \ P |. pipi:
aj < a < aj+1 pi j = 0, 1, ..., n 1, pi sQ pi sP pi pi Aj . pi ajAj
ajA1j + aA2j , pi A1j = [aj f < a] A2j = [a f < aj+1]. (5.28)
pi, pi Aj = A1j + A2j
sQ sP = ajA1j + aA2j ajAj = (a aj)A2j 0,
pi .
pi a > an, pi sQ pi sP pi pi An. pi anAn
anA1n + aA2n , pi A1n = [an f < a] A2n = [f a]. (5.29)
sQ sP = anA1n + aA2n anAn = (a an)A2n 0.
pi.
1 , pipi pi .
84
5.3.3. (X,A) f : X [0,] . pi pi - 0 s1 s2 ... f
sn f. pipi f , .
pi. Pn [0,]pi pi {sn} pi f Pn 0. ,
Pn =
{0 })
5.5. 89
15. fn : R R Lebesgue (n) n 0.
n=1
({x : fn(x) > n})
6
Lebesgue . pi (X,A, ) pi ( ) f : X R. pi pi :
(i) A A, A d = (A), pi A A.
(ii) : f, g a, b R
(af + bg) d = a
f d+ b
g d.
(iii) f f 0, fd 0. : f, g
f g, fd gd. :
(i) Lebesgue pi - (i) (ii) pipi.
(ii) pi pi pi.
(iii) pi f = f+ f .
pi pi pi Lebesgue pi pi pi- Lebesgue pi .
92
6.1 pi -
6.1.1. (X,A, ) f : X [0,] pi
f =
nj=1
aj(Aj) (6.1)
f pif d =
nj=1
aj(Aj), (6.2)
pi 0 = 0 = 0.
s
a1
a3a2
6.1: pi
6.1.2. () pi f d 0
A A : Ad = (A). (6.3)
() , f d = 0 ({x X : f(x) >
0}) = 0. pi pi-
pi :
6.1.3. (X,A, ) f : X [0,] pi pi :
f =
mj=1
bjBj (6.4)
pi b1, b2, ..., bm B1, B2, ..., Bm . :fd =
mj=1
bj(Bj). (6.5)
6.1. pi 93
pi. pi pi mj=1Bj = X (, -
Bm+1 = X \mj=1Bj bm+1 = 0 ). pi f
f =
ni=1
aiAi .
f pi , Ai Bj 6= ai = bj pipi :
Ai =
mj=1
(Ai Bj) Bj =ni=1
(Ai Bj), (6.6)
pi . , :fd =
ni=1
ai(Ai) =
ni=1
mj=1
ai(Ai Bj) =ni=1
mj=1
bj(Ai Bj) =
=
mj=1
ni=1
bj(Ai Bj) =mj=1
bj
ni=1
(Ai Bj) =mj=1
bj(Bj).
6.1.4. (X,A, ) , f, g : X [0,] pi a 0. :(i) :
af d = a
f d. (6.7)
(ii) pi:(f + g) d =
f d+
g d. (6.8)
(iii) :
f g X f d
g d. (6.9)
pi.
f =
ni=1
aiAi g =nj=1
bjBj
f g.
(i) af =ni=1 aaiAi af af d =
ni=1
aai(Ai) = a
ni=1
ai(Ai) = a
f d.
94
(ii) (Ai Bj)(i,j) pi pi
f + g =i,j
(ai + bj)AiBj (6.10)
(pi ) pipi (f + g) d =
i,j
(ai + bj)(Ai Bj) =
=i,j
ai(Ai Bj) +i,j
bj(Ai Bj) =
=
ni=1
ai
mj=1
(Ai Bj) +mj=1
bj
ni=1
(Ai Bj) =
=
ni=1
ai(Ai) +mj=1
bj(Bj) =
f d+
gd.
(iii) (g f) pi pi (ii):g d =
f d+
(g f) d
f d,
pi .
6.1.5. (i) (ii) , 6.1.3 pi pi Bj , : m
j=1
bjBj d =
mj=1
bj
Bj d =
mj=1
bj(Bj).
6.2
pi -. pi 5.3.3 f ,pi (sn) pi
sn f.
pipi, s pi 0 s f X, , pi ,
s d f d,
pi pi sd. pi
pi pi s pi (pi ) f - :
6.2. 95
6.2.1. (X,A, ) f : X [0,] . f pi
f d = sup
{s d : s pi 0 s f
}. (6.11)
, 6.1.1 pi, f pi supremum pi s = f pi pi ( 6.1.4(iii)).
A A, A
f d =
fA d, (6.12)
f A. Af d [0,] pi
X
f d =
f d.
6.2.2. (X,A, ) , f, g : X [0,] , A,B A a 0. :(i) :
af d = a
f d (6.13)
(ii) :
f g X f d
g d. (6.14)
(iii) A B A
f d B
f d. (6.15)
(iv) (A) = 0 f = 0 A, A
f d = 0. (6.16)
pi. (i) a = 0 pi. a > 0 , :af d = sup
{s d : s pi 0 s af
}=
= sup
{a
s
ad :
s
api 0 s
a f
}=
= a sup
{t d : t pi 0 t f
}= a
f d.
96
(ii) pipi pi pi pi : s pi 0 s f , 0 s g.
{s : s pi 0 s f} {s : s pi 0 s g} , pi
f d g d.
(iii) A B pi A B (;). , fA fB pi pi pi (ii).(iv) f = 0 A, fA = 0 X
fA d = 0,
Afd = 0.
(A) = 0 . s pi 0 s fA, s pi A pi
s =
ni=1
aiAi , pi Ai A i.
s d =
ni=1
ai(Ai) =
ni=1
ai 0 = 0.
pi s 0 s fA fA d =
Af d =
0.
pi pipi pi -, 6.2.1 pi pi pi , f, g : X [0,] , ,
(f + g) d =
f d+
g d. (6.17)
pi pi pi , Lebesgue Fatou. , pi 6.2.1: 6.2.3. (X,A, ) s : X [0,] pi . : A [0,]
(A) =
A
s d, (6.18)
A A 1 (X,A).pi.
s =
nj=1
ajAj
s, A A pi:
(A) =
A
s d =
sA d =
nj=1
aj
AAj d =
nj=1
aj(A Aj),
pi pi AB = AB . , j : A [0,] j(A) = (A Aj) , pi Aj , pi , aj 0 j.
1 s pi .
6.2. 97
pi pi lim infn : (An) - pi X,
lim infn
An = {x X : x An}. (6.19)
1.4 ( 4 1),
lim infn
An =
n=1
k=n
Ak. (6.20)
2.2 , pipi X (X,A, ), :
(lim infn
An) lim infn
(An). (6.21)
pi . -pi pipi pi :
6.2.4 ( Fatou). (X,A, ) fn : X [0,] . :
lim infn
fn d lim infn
fn d. (6.22)
pi. f = lim infn fn pi pi 5.2.4 pi s 0 s f X.
s d lim infn
fn d. (6.23)
, (0, 1)
s d lim inf
n
fn d.
(0, 1). An =
[fn s
]= {x X : fn(x) s(x)} (6.24)
pi pipi lim infnAn = X: , x X s(x) = 0 pi x An n, s(x) > 0,
s(x) < s(x) f(x) = lim infn
fn(x)
pi n0 n n0 s(x) < fn(x). , pix An, n n0 x lim infnAn.
pi An pi
fn(x) s(x)An(x) X (6.25) pi:
fn d sAn d =
An
s d = (An)
98
pi : A [0,] 6.2.3 pi s. pi pi (6.21) lim infnAn = X pi :
lim infn
fn d lim inf
n(An) (lim inf
nAn) = (X).
, (X) =Xs d pi
lim infn
fn d
s d,
pi 1 (6.23). , pi supremum pi pi s 0 s f pi .
6.2.5. (X,A, ) fn : X [0,] . fn f : X [0,] pipi fn f X n
f d = limn
fn d. (6.26)
pi. Fatou: lim infn fn = f :f d =
lim inf
nfn d lim inf
n
fn d
lim supn
fn d
f d,
fn f X. , pi
limn
fn d =
f d.
pi :
6.2.6 ( ). (X,A, ) fn : X [0,] . f = limn fn,
fn df d n. (6.27)
pi. pi pi pi , fn f X.
pi pi pi pi : f : X [0,] , pi 5.3.3 (sn) pi
sn f.
6.2. 99
, 6.2.6 f d = lim
n
sn d.
pi pi pi :
6.2.7. (X,A, ) f, g : X [0,] . :
(i) pi, (f + g) d =
f d+
g d. (6.28)
(ii) f g X pipi f d
100
(i) pi g : X [0,] fn g n g d
6.2. 101
6.2.10 (Beppo Levi). (X,A, ) fn :X [0,] . :
n=1
fn d =
n=1
fn d. (6.34)
pi. Sm = f1 + f2 + ...+ fm,
Sm f =n=1 fn :
Sm f m.pi, pi
n=1
fn d =
f d = lim
m
Sm d = lim
m
mn=1
fn d =
n=1
fn d,
pi pi (pipi) pi .
pi
(X,A, ) P (x) pi x X. P (x) pi
Z = {x X : P (x) } - ( 2.3.1). P .pi.. pi pi pi :
6.2.11. (X,A, ) f, g : X [0,] . :
(i) f = g .pi., f d = g d.(ii) f = 0 .pi. f d = 0.
pi. (i) X = {x X : f(x) 6= g(x)} pi Z A (- f, g ) pi pi (Z) = 0. pi, pi (iv) 6.2.2 pi:
f d =
X\Z
f d =
X\Z
g d =
g d.
(ii) f = 0 .pi. pi (i)f d =
0 d = 0.
pif d = 0 A = [f > 0] pi An =
[f 1n
]
A =
n=1
An.
102
0 =
f d
An
f d An
1
nd =
1
n(An),
(An) = 0 n. pi pi (A) = 0 pif = 0 .pi..
pi pi pi f .pi. f pi pi. , pi- pi 6.2.5 fn f .pi. fn f .pi.. pi - 6.2.8. pi pi .
6.2.1
pi 6.2.3. pi :
6.2.12. (X,A, ) f : X [0,] . : A [0,]
(A) =
A
f d (6.35)
A A. f pi .
:
6.2.13. (X,A, ) , f : X [0,] f pi . :
(i) .
(ii) A A (A) = 0 (A) = 02.(iii) g : X [0,] ,
g d =
gf d. (6.36)
pi. (i) pi () = 0. pi, (An) A (n=1An) =
n=1 (An),
n=1 An
f d =
n=1
An
f d. (6.37)
2 , pi pi .
6.2. 103
An , pi
n An
=
n=1
An
(6.37) n=1
fAn d =
n=1
fAn d
pi Beppo Levi fn := fAn .
(ii) pi (iv) 6.2.2.
(iii) pi pi , pi .
1. g g = A pi A A. pipi, pi
g d = (A) =
A
f d =
gf d.
2. g pi
g =
nj=1
ajAj .
pipi, pi 1 - :
g d =
nj=1
aj
Aj d =
nj=1
aj
Ajf d =
gf d.
3. g .
pipi, pi pi - (sn)n sn g. pi 2
sn d =
snf d
n , n pi pi g d =
gf d,
(sn) (snf) .
. pi 6.2.13 pipi : (X,A) pi (6.35); . Radon-Nikodym pi pi 10 pi .
104
6.3
: [,] . - pi pipi: f : X [,].
f = f+ f
pi f+ f . pi pi f , pi pi 6.2 pipi , :
f d =
f+ d
f d.
pi pipi pi . pi : 6.3.1. (X,A, ) f : X [,] .
(i) pi f+ d < f d < ,
f f d =
f+ d
f d. (6.38)
(ii) pipi , f+ d
6.3. 105
pi f d =
u d+ i
v d. (6.40)
pi, A A piA
f =
fA d. (6.41)
L1R() - 6.3.1 L1() 6.3.3. pi . pi pi - pipi L1(). pi pi L1R() pipi. 6.3.4. (X,A, ) , f, g L1() a, b C. :
(i) L1() , af + bg L1().(ii) L1() ,
(af + bg) d = a
f d+ b
g d. (6.42)
pi. (i) af + bg pi pi
|af + bg| |a||f |+ |b||g| pi
|af + bg| d |a||f | d+ |b|
|g| d
106
h d =
f d+
g d.
pipi, u1 = Ref , v1 = Imf , u2 = Reg v2 = Img. ,pi (6.40) pi pi :
f + g d =
(u1 + u2) d+ i
(v1 + v2) d =
=
(u1 d+ i
v1 d
)+
(u2 d+ i
v2 d
)=
f d+
g d.
: pi pi f pi R a R. , a 0
(af)+ = af+ (af) = af
pipi af d =
af+ d
af d = a
f d.
pi a < 0 pi :
(af)+ = af (af) = af+
(;) pi :af d = a
f d+ a
f+ d = a
f d.
f , u = Ref v = Imf a R
af d =
au d+ i
avd = a
f d
pi pi pi. a = i, af = v + iu f d =
v d+ i
u d = i
f d
pi pi (6.40). pipi , a = x+iy x, y R
af d =
(xf + iyf) d =
xf d+
iyf d =
= x
f d+ iy
f d = a
f d.
6.3. 107
6.3.5. f L1() f L1R() A,B A , :
ABf d =
A
f d+
B
g d. (6.43)
pi. A,B AB = A + B pi pi pi (ii) pipi pi.
pi pi pi - pi . pi L1R(). 6.3.6. (X,A, ) f, g L1R() f g.pi. X.
f d g d. (6.44)
pi. A = [f =] pi A
f d =
A
g d = (A).
pi 6.3.4 L1R() pi gAc fAc L1R() pipi:
Acg d =
Acf d+
Ac
(g f) d Acf d,
g f 0 .pi.. :g d =
A
g d+
Acg d
A
f d+
Acf d =
f d.
6.3.7. (X,A, ) f L1(). : f d |f | d. (6.45)pi.
f d = 0 pi. pipi
a C |a| = 1 f d = a f d (6.46) pi 6.2.4 f d = a f d = af d., pi pi pi (6.40) f d = Re(af) d |af | d = |f | d,pi .
108
pi : Lebesgue. pi Fatou pi .
6.3.8 ( ). (X,A, ) , fn : X C f : X C fn f .pi.. pi pipi pi g L1R() |fn| g .pi. X. fn f :
|fn f | d 0. (6.47)
pi pi
limn
fn d =
f d. (6.48)
pi. , |fn| g .pi. pi |fn| d
g d
6.4. 109
pi. pi pi M : pi
M d = M (X) t}).
F , pi limt+ F (t) = 0.
2.
[1,)1x d = 0.
3. {fn} pi pi : fn 0 limn
fn d = 1. pi pi {fn}
;
4. (X,A, ) . pi f fn, n N , fn f , pi k
fk < .
f d = lim
n
fn d.
110
5. (X,A, ) f : X [,] .pi f > 0 .pi.
Ef = 0 pi E,
(E) = 0.
6. f : R [0,] Lebesgue .
f d = limn
nn
f d = limn
{f1/n}
f d.
7. (X,A, ) f . -
f d = lim
n
{fn}
f d.
8. f . limx f(x) = 0;
9. fn = [n,n+1) Fatou pi .
10. {fn} (X,A, ).
lim supn
fn d
(lim supn
fn d
);
pi pi {fn} ;11. ( Chebyshev-Markov) (X,A, ) f : X
[0,] . t > 0
({x X : f(x) > t}) 1t
f d.
.
12. f fn, n N (X,A, ) fn f n N fn f .
f d = limn
fn d.
13. {fn} Lebesgue [a, b]. fn f , f
ba|fnf | d 0.
14. f, fn (X,A, ) fn f .pi pi
fn d
f d;
15. f, fn (X,A, ). |fn
f | d 0, fn d f d |fn| d |f | dmu.16. f, fn (X,A, ).
|fnf | d 0,
Efn d
Ef d E,
f+n df+ d.
6.4. 111
17. f (X,A, ). : > 0 pi E (E) <
E
f d >
f d .
pipi, E pi pi f E.
18. f (X,A, ). > 0 pi = () > 0 : (E) < ,
Ef d < .
19. f : R R Lebesgue . F (x) =
x f d .
20. (X,A, ) f, fn, n N fn f
limn
fn d =
f d 2k})
112
26. f : [a, b] R Lebesgue [a,x]
f d = 0,
x [a, b]. f = 0 .pi. [a, b].27.
limn
10
nx
1 + n2x2dx = 0 lim
n
10
n3/2x
1 + n2x2dx = 0.
.
28. pi ( pi )
n=0
pi/20
(1
sinx)n
cosxdx.
29. {fn}, {gn} g (X,A, ).pi |fn| gn, fn f , gn g ( pi) gn d
g d. f
fn d
f d.
30. f Lebesgue pi pipi [0, 1].
(i) Ef d = 0 E [0, 1] (E) = 1/2,
f = 0 .pi. [0, 1].(ii) f > 0 pi, inf{
Ef d : (E) = 1/2} > 0.
31. E Lebesgue pi Rk (E) 0 pi > 0 , A E Lebesgue (A) > ,
A
f d .
32. f L1[0, 1], 0. n fn(x) = f(xn) L1[0, 1].
33. (X,A, ) fn : X R
n=1
|fn| d < +.
:
(i) n=1 fn(x) x x.
(ii) n=1 fn (
n=1
fn d
)=
n=1
fn d.
6.4. 113
34. pi 0 < a < b fn(x) = aenax nenbx. n=1
0
|fn| =
0
( n=1
fn
)6=n=1
0
fn.
35. k, n N k n E1, . . . , En pi [0, 1] : x [0, 1] k pi E1, E2, . . . , En. pi i n (Ei) k/n.
36. {qn : n N} [0, 1] (an) pi
n |an|
114
(i) > 0 pi > 0 E [0, 1] Lebesgue (E) < ,
E|fn| d < n N.
(ii) pi (i) () pi
10|fn(t)| d(t) 1 n N.
41. (X,A, ) f . En = {x :|f(x)| n}, n (En) 0 n.
42. f : R R Lebesgue .(i)
Uf d = 0 U (U) = 1, f = 0
pi.
(ii) Gf d =
Gf d, G, f = 0
pi.
43. (X,A, ) f L1R(). pi pi C > 0
Ef d C E pipi-
. X
f d C.
pi pi f ;
44. A1, A2, ..., Ak, ... Lebesgue pi R :
() (Ak) 1/2, k () (Ak As) 1/4 k 6= s.
( k=1
Ak
) 1.
7
pi - pi . , X , fn : X R f : X C,
fn f fn(x) f(x) x X (7.1)
fn f fn f 0, (7.2)
> 0 pi n0() N
|fn(x) f(x)| < , n n0 x X.
- (X,A, ) . pi pi . , pipipi pipi - pi .
7.1
, pi pi pi pi- , pi 1. , :
7.1.1. (X,A, ) , fn : X C f : X C . (i) {fn} f .pi. pi Z A (Z) = 0
fn(x) f(x) x X \ Z.
116
(ii) {fn} f .pi. pi Z A (Z) = 0 fn f X \ Z,
sup{|fn(x) f(x)| : x X \ Z} 0. n. (7.3)
(iii) {fn} Cauchy .pi. pi Z A (Z) = 0 :
> 0 pi n0() N : m,n n0 x X \ Z |fn(x) fm(x)| < .
pi fn f .pi. fn f .pi.. pi, {fn} .pi. f , {fn} Cauchy .pi..
:
7.1.2. (X,A, ) fn : X C - . {fn} Cauchy .pi. pi f : X C fn f .pi..
pi. fn : A R Cauchy pi f : A R fn f A. pi X \ Z pi Z pi pi 7.1.1. pi .
pi pi :
7.1.3. (X,A, ) , fn : X C f, g : X C .
(i) fn f .pi. fn g .pi., f = g.pi..
(ii) fn f .pi. fn g .pi., f = g.pi..
pi. (i) pi .pi. , Z1, Z2 A (Z1) = (Z2) = 0
fn(x) f(x) X \ Z1 fn(x) g(x) X \ Z2.
, f(x) = g(x) x X \ (Z1 Z2). pi pi pi (Z1 Z2) = 0.(ii) pi (i) .pi. pi .pi. .
7.1.4. (X,A, ) , fn, gn : X C f, g : X C .
(i) fn f .pi. gn g .pi., a, b R afn + bgn af + bg .pi..
7.1. 117
(ii) fn f .pi. gn g .pi., a, b R afn + bgn af + bg .pi..
pi. (i) pi pi, Z1, Z2 A (Z1) = (Z2) = 0
fn(x) f(x) X \ Z1 gn(x) g(x) X \ Z2., x X\(Z1Z2) afn(x)+bgn(x) af(x)+bg(x), pi (Z1Z2) =0. pi afn + bgn af + bg .pi..(ii) pi Z1, Z2 A
fn f X \ Z1 gn g X \ Z2., pi afn + bgn af + bg - X \ (Z1Z2). pi pi (Z1 Z2) = 0.
7.1.5. (X,A, ) , fn, gn : X C , f, g : X C a, b C.(i) fn f .pi. gn g .pi., fngn
fg .pi..(ii) fn f .pi., gn g .pi pipi, pi
M > 0 |fn| M |gn| M .pi. n, fngn fg .pi..
pi. (i) pi (i) pi .
(ii) .pi. pi. .pi. , - Z1, Z2 A (Z1) = (Z2) = 0
fn f X \ Z1 gn g X \ Z2.pi pi, n pipi An A (An) =0 |fn| M |gn| M X \An.
Z = Z1 Z2 n=1
An (7.4)
pi Z A pipi
(Z) (Z1) + (Z2) +n=1
(An) = 0,
(Z) = 0.
> 0. x X \ Z |fn(x)gn(x) f(x)g(x)| =
(fn(x)gn(x) f(x)gn(x))+ (f(x)gn(x) f(x)g(x)) |fn(x) f(x)||gn(x)|+ |f(x)||gn(x) g(x)| M(|fn(x) f(x)|+ |gn(x) g(x)|).
118
|f(x)| M X \ Z (;). , pi, N N x X \ Z n N
|fn(x) f(x)| < 2M
|gn(x) g(x)| < 2M
.
pi, pipi, x X \ Z n N
|fn(x)gn(x) f(x)g(x)| M M
= ,
fngn fg .pi. pi .
. {fn} {gn} pi- pi pi pi. pi.
7.2
pi pi pi (X,A, ). . f : X C pi, pi
E[f ] =f d (7.5)
f , pi pi, f . :
7.2.1. (X,A, ) , fn : X C f : X C . :(i) {fn} f
|fn f | d 0. (7.6)
(ii) {fn} Cauchy > 0 pi n0() N : m,n n0
|fn fm| d < . (7.7)
pi , {fn} f , Cauchy .
pi 7.1 pi :
7.2.2. (X,A, ) , fn : X C f, g : X C . fn f fn g , f = g .pi..
7.2. 119
pi. n
|f g| d |f fn| d+
|fn g| d 0
pi . , |f g| 0, pi |f g| = 0.pi. f = g .pi.. 7.2.3. (X,A, ) , fn, gn : X C f, g : X C . fn f gn g , a, b C afn+bgn af+bg .
pi. pi pi : (afn + bgn) (af + bg) d |a| |fn f | d+ |b| |gn g| d 0. pi Cauchy
pi:
7.2.4 (Riesz). (X,A, ) fn : X C . {fn} Cauchy , pi f : X C fn f . pipi,pi pi {fnk} {fn} fnk f .pi..pi. f . {fn} Cau-chy , k pi nk N m,n nk
|fn fm| d < 12k.
pi pi n1 < n2 < ... (;) pi {fnk} pi {fn}. pi pi pipi
|fnk+1 fnk | d 0 |fn| M |gn| M .pi. n N, fngn fg .
pi. (i) pi , pi pi {fnk} {fn} pi f .pi. pi pi 7.1.5 (ii).
(ii) pi 7.1.5 (ii) pi Z A (Z) = 0 x X \ Z n N
|fn(x)| M |gn(x)| M |g(x)| M pi (i). , : fngn fg d = (fngn fng) + (fng fg) d
|fn||gn g| d+
|fn f ||g| d
M|gn g| d+M
|fn f | d 0.
pi, pi fngn fg .
7.3
7.3.1. (X,A, ) , fn : X C f : X C . :(i) {fn} f ( pi), > 0
({x X : |fn(x) f(x)| }) 0. (7.10)
(ii) {fn} Cauchy , > 0 pi n0(, ) N: m,n n0
({x X : |fn(x) fm(x)| }) < . (7.11)
pi pi :
7.3.2. f, g : X C , a, b > 0 :
({x : |f(x) + g(x)| a+ b}) ({x : |f(x)| a})+ ({x : |g(x)| b}).
pi .
122
pi pipi , {fn} f , {fn} Cauchy pi.
7.3.3. (X,A, ) , fn : X C f, g : X C . fn f fn g , f = g .pi..pi. > 0. , pi pi pi pi, n :
({x : |f(x)g(x)| }) ({x : |f(x)fn(x)|
2})+({x : |fn(x)g(x)|
2})
pi n. ({x X : |f(x)g(x)| }) = 0 > 0 pi f = g .pi. (;). 7.3.4. (X,A, ) , fn, gn : X C f, g : X C . fn f gn g , a, b C afn + bgn af + bg .
pi. pi pi a 6= 0 b 6= 0. pi pi :
({x : |(afn(x) + bgn(x)) (af(x) + bg(x))| })
({x : |fn(x) f(x)| 2|a| }
)+
({x : |gn(x) g(x)| 2|b| }
) 0 n. pi pi .
pi {fn} pi Cau-chy . lim supn : (An) pi X
lim supn
An = {x X : x pi pi pi An}. (7.12)
1.4,
lim supn
An =
n=1
k=n
Ak. (7.13)
lim supn pi-:
7.3.5 (1 Borel-Cantelli). (X,A, ) (An) A.
n=1
(An)
7.3. 123
2.3. pi pipi pipi .
pi pi pi :
7.3.6. (X,A, ) fn : X C - . {fn} Cauchy , pi f : X C fn f . pipi pipi {fnk} {fn} fnk f .pi..pi. {fn} Cauchy , k nk N
({x X : |fn(x) fm(x)| 1
2k
}) 0 |fn| M |gn| M .pi. n N, fngn fg .
7.4. 125
pi. (i) pi pi pi {fnk} {fn} pi f .pi. pi pi 7.1.5.(ii) , , Z A (Z) = 0
|fn(x)| M |gn(x)| M
x X \ Z. > 0. :
({x : |fn(x)gn(x) f(x)g(x)| })
({x : |fn(x) f(x)| 2M}) + ({x : |gn(x) g(x)|
2M})
( pi) pi .
7.4
pi pi - .pi. . :
7.4.1. (X,A, ) , fn : X C f : X C . :
(i) {fn} f > 0 pi A A (A) < fn f X \A.
(ii) {fn} Cauchy > 0 pi A A (A) < {fn} Cauchy X \A.
pi pi fn f {fn} Cauchy .
7.4.2. (X,A, ) , fn : X C f, g : X C . fn f fn g f = g .pi..
pi. > 0 E = {x X : f(x) 6= g(x)}. pi, A1, A2 A
fn f X \A1 fn g X \A2 (A1), (A2) < . , X \ (A1 A2) f = g (;) E A1 A2. pi
(E) (A1) + (A2) < 2
> 0 , pi f = g .pi..
7.4.3. (X,A, ) , fn, gn : X C f, g : X C . , a, b C afn + bgn af + bg .
126
pi. > 0. pi A1, A2 A (A1), (A2) 0 k
1k < . fn f X \ Ak (Ak) < 1k < pi.
7.4.5. (X,A, ) , fn : X C f : X C . fn f , fn f .pi..pi. pi pi {fn} pipiCauchy .
pi pi, 7.4.4 :
7.4.6. (X,A, ) , fn, gn : X C f, g : X C .(i) fn f pipi pi M > 0 |fn| M
.pi. n N, |f | M .pi..
7.5. 127
(ii) fn f pipi |fn| M |gn| M .pi. n N, fngn fg .
pi. (i) pi pi fn f .pi. pi pi 7.1.5 (ii).
(ii) > 0. pi A1, A2 A (A1), (A2) < /2 pipi
fn f X \A1 gn g X \A2.
pi 7.1.5 (ii) pi fngn fg .pi. X \ (A1 A2) (A1 A2) < pi .
7.5
pi pi pi . pi : 7.4.4 {fn} f f .pi.. : pi, Egorov, 2 pi 3 Littlewood pi 4.
7.5.1 (Egorov). (X,A, ) , fn : X C f : X C . (X) 0. pi pi fn f pi X ( ). k,m N
Ak,m =
{x X : |fn(x) f(x)| < 1
k n m
}(7.16)
pi (Ak,m)m=1 (;). pipi, pi- x X k N, pi m N |fn(x) f(x)| < 1/k n m. pi
X =
m=1
Ak,m. (7.17)
pi (Ak,m) (X) m k pi mk N
(X) < (Ak,mk) +
2k.
A =
k=1
Ak,mk .
128
fn f A: > 0. k N 1/k < . x A x Ak,mk n mk
|fn(x) f(x)| < 1k< ,
(fn f)|A < .pi,
(X \A) k=1
(X \Ak,mk) 12k}. (7.18)
. k ({x X : g(x) > 1/2k}) g d
Sk
g d >
Sk
1
2kd =
1
2k(Sk) = :
pi.
pi pi, (X \Ak,m)m=1 , , limn (X\Ak,m) = 0 k. , mk N
(X \Ak,mk) 0. > 0 pi A A (A) < fn f X \ A. pi , n0 N x X \A n n0
|fn(x) f(x)| < .
pi{x X : |fn(x) f(x)| } A
({x X : |fn(x) f(x)| }) (A) <
n n0. pi fn f . , pi 7.3.6:
pi, (Fm) < 12m1 m. > 0. pi m 12m1 < x X \Fm
|fnk(x) f(x)| 0. pi Chebyshev-Markov
({x X : |fn(x) f(x)| }) 1
|fn f | d 0,
fn f . , pi fn f . , pi fn f .
, pi 0 > 0 pi {fnk} {fn} |fnk f | d 0 (7.20)
k. fn f fnk f , pi 7.3.7, pi pi fnkl fnk fnkl f .pi.. pi |fnkl | g g 6.3.8
|fnkl f | d 0
pi (7.20). , pi fn f , pi .
7.5.7. , , -pi .
7.5. 131
pi. fn : (0, 1) R fn = n(0, 1n ). , (0, 1]
({x X : |fn(x) 0| }) = 1n 0
fn 0 . , n |fn(x)| dx = n 1
n= 19 0
{fn} 1. -
pipi :
7.5.8. (X,A, ) , fn : X C f : X C . pi (X) 0 pi {fnk} {fn} |fnk f |
1 + |fnk f |d (7.21)
k. , pi (ii) pi pi {fnkl } {fnk} fnkl f .pi. |fnkl f |
1 + |fnkl f | 0.
pipi |fnkl f |1 + |fnkl f | 1
X pi pi 6.3.9 |fnkl f |1 + |fnkl f |
d 0
pi (7.21).
1 pi ;
132
(iii) (i) > 0. pi Chebyshev-Markov pi:
({x : |fn(x) f(x)| }) = ({
x :|fn(x) f(x)|
1 + |fn(x) f(x)|
1 +
})
1 +
|fn f |1 + |fn f | d 0,
(iii).
pi pi pi:
-
- pi
()
() ()
() ()()
()
1:
1
7.2:
pi:(): pi(): (X)
7.6. 133
2. {An} f , pi A A f = A .pi.
3. pi (X) < fn : X C supn |fn(x)| 0 pi B A (X \B) <
supnN,xB
|fn(x)| 0
limn
( k=n
Ek()
)= 0.
8. pi (X)
134
11. pi (X)
8
pi :
1. , f : X Y , pi (X,A) (Y,B) .
2. pi pi pi - .
3. Lebesgue Riemann pi pi pi pi .
pi pi - Lebesgue.
8.1 pi
5.1.7 (iv) pi (X,A) f : X R B B(R) f1(B) A. pi pi -:
8.1.1. (X,A) (Y,B) f : X Y . f (A,B)- ( pi A B) B B f1(B) A., pipi pi Y B = B(Y ) f A-.
f1(B) = {f1(B) : B B} (8.1)
136
pi f (A,B)- f1(B) A. (8.2)
8.1.2. (X,A), (Y,B) (Z, C) f : X Y g : Y Z. f (A,B)- g (B, C)-, g f (A, C)-.pi.
(g f)1(C) = f1(g1(C)). (8.3)pi pi pipi f1(B) A g1(C) B. pipi :
f1(g1(C)) f1(B) A
pi (8.3) .
pi pi pi f : (X,A) (Y,B) (ii) : 8.1.3. (X,A) (Y,B) f : X Y . :
(i) F = {B Y : f1(B) A} - X.(ii) C P(Y ) (C) = B, f (A,B)-
f1(C) A. (8.4)pi. (i) pi - . (ii) , f
B {B Y : f1(B) A}. (8.5) F - (C) = B C F B F . 8.1.4. () (i) pi f1(B) - X.()
= {(, b] : b R} () = B(R) pi (ii) 5.1.1 f : (X,A) R.
pi , pipi (X,A) . , f pi (Y,B) : 8.1.5. (X,A) (Y,B) f : X Y (A,B)- . (X,A), : B [0,] pi
(B) = (f1(B)
):= (f B), (8.6)
B B. pi (pi ) (Y,B). f f() f .
8.1. pi 137
Lebesgue. , pi pi pi f() pi.
8.1.6. (X,A) (Y,B) f : X Y (A,B)- . (X,A) g : Y [,] g : Y C ,
B
g df() =f1(B)
g f d, (8.7)
B B. ( pi pi pi pipi .)
pi. , 8.1.2 g f pi . pipi, pi
(g f) f1(B) = (g B) f (8.8)pi pi B = Y ( g = g B), f1(B) = X. = f() pi
g d =
g f d.
pi pi , :
1. g g = B pi B B.pi :
g d =
Bd = (B) =
(f1(B)
)
g f d =B f d =
f1(B) d =
(f1(B)
) .
2. g pi
g =
mj=1
bjBj .
pi 1 :g d =
mj=1
bj
Bj d =
mj=1
bj
Bj f d =
g f d,
pi .
3. g .
138
pi 5.3.3, pi ,pi (sn)n sn g Y . pi 2, n
sn d =
sn f d.
(sn)n (sn f)n sn g sn f g f . 6.2.6 pi, pi
g d = limn
sn d = lim
n
sn f d =
g f d.
4. g : Y [,] ., , g = g+ g pi , pi 3
g+ d =
g+ f d =
(g f)+ d
g d =
g f d =
(g f) d.
, pi g d pi pi
g f d
pipi .
5. g : Y C . 4 pi u = Ref v = Imf .
8.2 Luzin
pi Rk pi . pi pi pi , pi pi, Luzin, pi pi 3 Littlewood pi .
8.2.1 (Luzin). A Rk Lebesgue (A) < f : A R . > 0 pi F A (A \ F) < f |F .
pi. > 0. pi pi .
1. f f = E pi E A Lebesgue . pi f pi 1 0. pi Lebesgue, pi K A G A K E G
(E) (K) + 4, (G) (E) +
4.
8.2. Luzin 139
A
E
K
G
8.1: pi Luzin
(E) (A)
140
(A \An) < 2n+2 sn|An .
A =n=1
An
pi
(A \A) n=1
(A \An) 0 pi F A (A \ F) < F f . pipi f pi A (;) pi, pi pi pi, f Riemann . pi pi f F .
pi, f = Q [0, 1] [0, 1] pi f (R\Q)[0, 1] 0.
8.3 Riemann
f : [a, b] R baf(x) dx Rie-
mann baf d Lebesgue f ( pi). pi
pi pi. pi pi , Lebesgue pi Riemann.
8.3.1. f : [a, b] R Riemann . ,(i) f .
(ii) f Lebesgue ba
f d =
ba
f(x) dx. (8.10)
8.3. Riemann 141
pi. pi :
1. .
2. h 0 Eh d = 0, h = 0 pi E.
pi, f g Ef d =
Eg d, f = g pi E.
3. s =i[ai,bi] , b
a
s d =
ba
s(x) dx.
pi f Riemann . , pi (Pn) [a, b] : Pn Pn+1 ( Pn+1 pi Pn), Pn 0 ( pi Pn 0),
L(f, Pn) ba
f(x) dx , U(f, Pn) ba
f(x) dx.
`n ba`n(x) dx = L(f, Pn) (, L(f, Pn) =k1
i=0 mi(xi+1 xi) `n =k1i=0 mi[xi,xi+1)) un
baun(x) dx = U(f, Pn). ,
`n f un.pi Pn Pn+1 pi (`n) (un) , pi ` = limn `n u = limn un ` f u. pi , b
a
u d = limn
ba
un d = limn
ba
un(x) dx =
ba
f(x) dx
ba
` d = limn
ba
`n d = limn
ba
`n(x) dx =
ba
f(x) dx.
` u ba` d =
bau d, pi ` = u pi.
` f u, pipi ` = f = u pi. (8.11)
, f ( pi) ( pi). pi (i).
f , f Lebesgue . ,pi (8.11) b
a
f d =
ba
u d =
ba
f(x) dx,
pi (ii).
pi Riemann - f : [a, b] R: pi pi.
142
8.3.2. f : [a, b] R . f Riemann
({x [a, b] : f x}) = 0.
pi. pi pi f pi. pi - (Pn) [a, b] Pn Pn+1, Pn 0, U(f, Pn) L(f, Pn) 0.
`n, un pi Pn, `n f un, ba`n(x) dx = L(f, Pn)
baun(x) dx = U(f, Pn). , Pn = {a = x0 0. f x,pi > 0 : y, z (x , x+ ) |f(y) f(z)| < . pi n0 pi Pn0 < . [xi, xi+1] pi Pn0 pi x, [xi, xi+1] (x , x+ ),
Mi mi = sup{f(y) : y [xi, xi+1]} inf{f(z) : z [xi, xi+1]} ,
0 un0(x) `n0(x) . ,
0 u(x) `(x) un0(x) `n0(x) .
> 0 , pi u(x) = `(x). (A P ) = 0 ` = u pi, pi
ba` d =
bau d.
: pi f Riemann [a, b]. pi- (Pn)n Pn Pn+1 n
L(f, Pn) ba
f(x) dx , U(f, Pn) ba
f(x) dx
8.4. 143
n N, `n un pi Pn, `n f un b
a
`n(x) dx = L(f, Pn) ,
ba
un(x) dx = U(f, Pn).
(`n) (un) . ` = limn `n u = limn un. ` f u pi b
a
` d = limn
ba
`n(x) dx = limnL(f, Pn) =
ba
f(x) dx
ba
u d = limn
ba
un(x) dx = limnU(f, Pn) =
ba
f(x) dx.
, ba
` d =
ba
u d. (8.14)
` u, pi ` = u pi. C = {x [a, b] : `(x) = u(x)} P = n=1 Pn.
x C \ P f x. : x C \ P > 0. `(x) = u(x), pi n0 0 un0(x) `n0(x) < . (xi, xi+1) pi Pn0 pi x,
sup{f(y) : y [xi, xi+1]} inf{f(z) : z [xi, xi+1]} < .
pi f x ( ).
pi A f , A ([a, b] \ C) P , (A) = 0.
. pi pi pi pi Rk .
8.4
.
1. X,Y Borel pi- . pi f : X Y .
2. f, g : R R f(x) = ex g(y) = y3+y. pi
(
Recommended