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

(