Σπηλιώτης-Θεωρία Μέτρου σε απειροδιάστατους χώρους.pdf

ANTI PROLOGOU

pi pi pi () pi pi pi pipi, pi , . -. , pi pi, pi pi pi. pipi pi pipi 200 pi pi pi . pi pi . pi . pi : pi - pi (pi..tightnes) pi pipi pi. - pipi pi pi. pi completelyregular ( pi pi. pi completelynormal pi pi ). pi pi - pi.

2010

. pi

i

PINAKAS SUMBOLWN

(E ) - pi pi pi E

() - pi pi Y Xpi (Y,H ) .

FE - F .

T TX T (X) pi. X.G GX G (X) pi. X.

K KX K(X) pi pi. X.B BX B(X) - Borel pi. X (B = (T )).B(X, ) - Borel pi pi X.

Bm - Borel Rm.

C(X,) RX pi pi .

C(X,) RX - pi pi - , (C(X,)).

C(X) C(X,R) pi pi. X.

Cb(X) pi pi. X.

K(X) pi pi - pi. X.

(X,X ) pi pi. pi. . X.

C (X,X ) pi pi. pi. . X.

(X,X ) pi Mackey pi. pi. . X.

S(X,X) pi Sazonov pi. pi. . X.

ii

Perieqmena

1 - 31.1 . . . . . . . . . . . . . . . . . . . 31.2 Rm. Lebesgue . . . . . . . . . . . . . . . . 111.3 pi . . . . . . . . . . . . . . . . . . . . . . . 141.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 pi Hausdorff 192.1 pi . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 pi . . . . . . . . . . . . . . 222.3 pi pi pi . . . . . . . . . 252.4 . . . . . . . . . . . . . . . . . . . . 272.5 pi . . . . . . . . . . . . . 372.6 . . . . . 402.7 pi . . . . . . . . . . . . . . . . . . . . . . 432.8 . . . . . . . 462.9 . . . . . . . . . . . . . . . . . 472.10 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 pi pi 553.1 pi pi . . . . . . . . . . . . . . 553.2 Prohorov . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3 Kolmogorov . . . . . . . . . . . . . . . . . . . . . . . . 603.4 pi -

tTB(St) T pi 66

3.5 Ionescu-Tulcea . . . . . . . . . . . . . . . . . . . . . . . 71

4 pi pi. . 754.1 pi pi. . . . . . . . . . . 754.2 - - pi. . . . . 78

4.2.1 . . . . . . . . . . . . . . . 784.2.2 - pi. . . . . . . . . . 79

4.3 - . . . . . . . . . . . . . . 854.4 pi ... Prohorov 85

1

5 pi 915.1 . . . . 915.2 . . . . . . . . 95

6 Minlos Sazonov 996.1 . . . . . . . . . . . . . . . . . . . . . . . . . 996.2 pi Sazonov . . . . . . . . . . . . . . . . . . . . . . . . . 1026.3 Minlos Sazonov . . . . . . . . . . . 1076.4 pi .. . . . . . . . . . . 114

7 Gauss 1177.1 Gauss Rn . . . . . . . . . . . . . . . . . 1177.2 pi Gauss pi -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.3 . . Gauss Hilbert.

Mourier . . . . . . . . . . . . . . . . . . . . . . . . . . 122

pi 2.4.2. 125

pi 2.4.4. 129

pi Prohorov 135

2

Keflaio 1

Mtra kai s-lgebrec

1.1 Mtra se afhrhmnouc qrouc

6= 1.1.1. F pi - pi pi pi:

1. F2. A F Ac F

3. (An, n N) F n=1

An F

pi pi :

F A,B F A B, A B, A \B, A M B F

(An, n N) F n=1

An F

1.1.2. ( -)

1. F = {,}2. F = {,, A,Ac} pi A A 6= ,

3. =ki=1

Ai Ai Aj = i 6= j {1, ...k}

F =

{iI

Ai : I {1, ..., k}}

- cardF = 2k

4. F = P() pi .

3

1.1.3. {Fi, i I} - pi F =

iIFi -. ( F

, Fi i I.) 1.1.4.

iIFi.

pipi :

1.1.5. C pi . - pi pi C - pi - pi pi C (pi , P()). - pi pi C (C ) pipi pi pi:

(C ) C - F C F (C )

(C ) ( ) -pi pi C .

1.1.6.

1. C1 C2 (C1) (C2)2. C - (C ) = C

3. C E (C ) (E ) = (C ) pi - - Borel Rm pi :

1.1.7. E pi Rm. - - Borel pi Rm - Bm (E ). pi ( [2])

1.1.8. Pm = {(a1, b1] ... (am, bm] : ai < bi R}{} H pi Rm

Bm = (H ) = (Pm).

Bm = (L )pi L = {B(x, r) : x Rm, r > 0} , B(x, r) = {x Rm : |x| < r} 1.1.9. , pi pi Rm Bm. pi pi pi Rm. :

x Rm {x} Bm ( ) R Q B1 Q

Q =xQ{x}.

4

: - pi pi C pi . : pi C ; pi , (transfinite in-duction) pi [1] [9] . pi - Borel pipi pi :

cardBm = cardR c 1.1.10. (,F ) pi 6= F - pi . (,F ) pi

: F 7 [0,]pi pi pi pi:

1. () = 02. An F , n N Ai Aj = i 6= j N

( n=1

An

)=

n=1

(An).

pipi () < + -pipi pi En F , n N =

n=1

En

(En) < +. 1.1.11. () (,F ). :

1. A B (A) (B)2. A B (A) < + (B \A) = (B) (A)

3. An F , n N (n=1

An) n=1

(An)

4. An F , n N An An+1n N

( n=1

An

)= limn(An)

5. An F , n N An An+1 n N (Ak) < + pik N

(

n=1

An) = limn(An)

5

1.1.12. 1, 2 (,F )

1(A) = 2(A) A F . pi pi -. pi pi Dynkin Dynkin . pi [7] . pi pi pi . ( [6])

1.1.13. 1, 2 (,F ) F = (C ) pi C pi pi pi:

A,B C A B C

pi En C , n N En En+1 n N n=1

En =

1(A) = 2(A) A C 1(Ek) = 2(Ek)

N = { : pi N F (N) = 0 N}F = {A N : A F N N}

: F 7 [0,] : (A N) = (A). F - pi pi F , (, F ) pi pi (, F , ) pi. pi pi (, F , ) minimal ( (,F1, 1) pi F1 F 1|F = F1 F 1|F = ). 1.1.17. pi F = (F N). F = {A pi B1, B2 F B1 A B2 (B1) = (B2)}

pi. [6]

pi -pi (,F ) C F . pi .

1.1.18. 6= P() pi . : P() 7 [0,] pipi pi:

1. () = 02. A B (A) (B)

3. An , n N (n=1

An) n=1

(An)

pi pi:

1.1.19. pi . pi A -

(A E) + (Ac E) = (E) E (?)

- pi M ,

M = {A : (A E) + (Ac E) = (E) E }

1.1.20. ?

(A E) + (Ac E) (E) E

pi : (B) = 0 B M pi B (B) = 0 M

7

1.1.21. () piM - pi - M , (,M , ) pi.

pi. [3] [6] [7] [9] ...

:

1.1.22. ( ) C pi C . 0 : C 7 [0,] pi 0() = 0. : P() 7 [0,] :

(A) = inf

{ n=1

0(Bn) : Bn C n=1

B A}

: inf = + pipi (C , 0)

pi. [11]

- pi . pi pipi pi . pi pi ( ).

1.1.23. C pi - pi pi pipi:

1. C

2. A,B C A B C

3. A,B C A B pi E1, E2, ..., Ek C B \A =

ki=1

Ei

pipi C C

1.1.24.

1. = Rm Pm = {(a1, b1] ... (am, bm] : ai < bi R} {} Pm ( m = 1).

8

2. (1,F1) (2,F2) pi F1,F2 pi 1,2 . :

C = {AB : A F1B F2}

. pi F1,F2 .

pi pi pi pi 0 : C 7 [0,] piC pi pi pi:

() 0() = 0

() An C , n N Ai Aj = i 6= j n=1

An C

0(n=1

An) =n=1

0(An).

(), ().

1.1.25. (pi) C pi 0 : C 7 [0,] pi (),() pipi

() pi En C , n N En En+1 n=1

En =

0(En) < + n N. :

1. pi (, (C )) pi |C = 0 ( pi 0 - (C )). -pipi.

2. , pi pi (C , 0) M (C ) |(C ) =

3. (, ((C ), ) pi (, (C ), ) (C ) =M =

1.1.26. ( pi) pi () pi pi

... limn0(En) = 1.

pi -pipi pi (1) pi pi, () = 1. pipi pi .

9

pi. [8] [9] [11]

1.1.27.

1. pi pipi

M = {A : (A) + ( \A) = 1}

( M 7)

2. C (. C ) 1.1.26 () pi pi : 0() = 1.

1. A (A) < +. pi B (C ) pi A B (A) = (B). (C ) B \ A () = 0.

2. F0 pi , pi pi pi pi:

1. F02. A F0 Ac F03. , F0 A B F0.

0 : F0 7 [0, 1] pi pi pi:

i. 0() = 1

ii. A1, ..., Ak F0 0(A1 ...Ak) = 0(A1) + ...+0(Ak)

iii. An F0, n N An An+1 n=1

An = limn0(An) = 0

(F0, 0) pi pipi 1.1.26pi pipi pi pi 2 pipi.

3. C pi 0 : C 7 [0,]pi pipi:

() 0() = 0

() k N A1, A2, ..., Ak C ki=1

Ai C 0(A1 ...Ak) =0(A1) + ...+ 0(Ak)

10

(B) An C , n N n=1

An C 0(n=1

An) n=1

0(An)

() () (B) () () ( pi pi pi pi ( pi) (),(),(B) (),().(Updeixh: Prta dexte epagwgik to akloujo Lmma: gia kje k N kaiA1, ..., Ak C xna metax touc kai B C me B

ki=1

Ai isqei B \ki=1

Ai =l

j=1

Ej pou Ej Cxna metax touc.)

4. (,F , ) . F -, : F 7 [0,]. pi pi (F , ).( 8). (A) = inf{(B) : B F B A}, A .

pi pi pi . pi pi .

1.1.28. (,F ) ,. F -pi . E . pi pi FE = {A E : A F} - F = (C ) FE = (CE) pi CE = {B E : B C }. - FE F E.

pi. [9] 132

pi ( [6] 164 [7])

1.1.29. (,F , P ) pi P pi pi (F , P ).pi pi E

P (E) = 1 (4) - FE = {A E : A F} P0 : FE 7 [0, 1] :

P0(A E) = P (A). (E,FE , P0) pi.( E F ). 1.1.30. pipi (,F , )pi () < +. pi (4) pi (E) = () ( E F (E) = ().

1.2 Mtra ston Rm. To mtro Lebesgue Rm pi

Pm = {(a1, b1] ... (am, bm] : ai < bi R} {}

11

Bm = ((Pm)). 0 : Pm 7 [0,) :

0() = 00((a1, b1] ... (am, bm]) =

mi=1

(bi ai)

pi pipi (),(),() 1.2.25 0 C = Pm. () () pi.( pi En = (n, n] ... (n, n], n N). pi () pi- ( pi. [2] [7] [9]). pipi (Pm, 0) pi:

1.2.1. pi (Rm,Bm) pi

((a1, b1] ... (am, bm]) =mi=1

(bi ai).piM Bm (A) = (A) A Bm. (Rm, Bm, ) pi (Rm,Bm, ) Bm =M = . = (Rm, Bm) = (Rm,M) Lebesgue Rm.

pi. [12]

Lebesgue Rm pi:

1.2.2.

1. (K) < + pi K Rm

2. A M (A) = inf{(U) : U Rm U A} pi > 0 pi pi U Rm pi: U A (U \A) < .

3. A M (A) = sup{(K) : K pi pi A} (A) < > 0 pi pi K Rmpi K A (A \K) < .

4. A M pi F- G- pi E A H (E) = (A) = (H). (H \ E) = 0.

pipi Lebesgue pi Rm. pi .

1.2.3.

1. A Rm x Rm :A M(Bm) A+ x M(Bm).

2. (A+ x) = (A) A M x Rm.

12

3. (Rm,Bm) (K) < + pi K Rm (I + x) = (I) I Rm x Rm pi a 0 pi (A) = a(A),A Bm.

Lebesgue (Rm,Bm) (pi pipi ) Haar pi ( )pi pi pi pi (Rm,+).

pi pi R. - :

1.2.4. F : R 7 R ,

limt 7F (t) = 0 , limt 7+F (t) = 1.

F .. P0 : P1 7 [0, 1] :P0() = 0 P0((a, b]) = F (b) F (a)

pi (P1, P0) pi pi 1.1.26, (),() pi (). pi:

1.2.5. .. F , pi pi- P (R,B1) pi P ((a, b]) = F (b)F (a) a < b R.

1.2.6. pi ,: pi P (R,B1) pi .. pi P ((a, b]) = F (b) F (a). F (t) = P ((, t]), t R.

P ((a, b]) = F (b)F (a) pi (R,B) R. pi R .

cardB1 = c. pi cardP(R) = 2c c < 2c pi B1 6= P(R).pi M = P(R) (;) pi . :

1.2.7. pi pi A (0, 1) pi A /M .pi. [12]

, BanachKuratowski

1.2.8. (Banach-Kuratowski) pi pi I = [0, 1] (.(I,P(I))) pi pi pi (I) = 1 ({x}) = 0 x I.

13

pi. [10]

pi pi pipi pi pi ( ) pi pi - .

pi pi pi - ; Solovay 1965 pi : pi: pi R ,M = P(R) consistent Zermelo-Frankel pi

cardM . pi : Cantor B = [0, 1] \ ( 13 , 23 ) \ ( 19 , 29 ) \ ( 79 , 89 ) \ ... B =

{ n=1

an3n : an = 0 2

}. pi :

(B) = 0 cardB = c.

pi pi Lebesgue M P(B) M . card(B) = c cardP(B) = 2c pi cardM = 2c. cardB1 = c pi pi M /B1. pi pi Suslin sets .

1.3 Metrsimec Apeikonseic

1.3.1. (X,F ) (Y,H ). pif : X 7 Y F H f1(B) F B H . 1.3.2. pi pi H = (E ).: f F H f1(B) F B E .pi. A = {B H : f1(B) F} pi - A E A H . 1.3.3. pi pi Y d H = (E ) pi E ( H - Borel ). F H pi fn : X 7 Y , n N pi f : X 7 Y :f(x) =

d

limnfn(x) x X.

f : X 7 Y F H .

14

pi. (): U E Uk = {y Y : d(y, U c) > 1k}, k = 1, 2, ....U =

k=1

Uk pi f1(U) =k=1

f1(Uk).

f1(Uk) =m=1

nm

f1n (Uk) Uk E .

pi pi .

1.3.4. X 6= (Y,H ) . Y X - pi X pi Y . - pi pi

() = ({f1(A) : f , A H }) - ( ) pi . pi H = (E )

() = ({f1(A) : f , A E })

1.4 Mtro Ginmeno

(1,F1, 1) (2,F2, 2) pi 1, 2 -pipi.:

C = {AB : A F1, B F2} pi C pi 1 2( pi pi 12). - F1,F2 - (C ). - F1F2. :

F1 F2 (C ) = ({AB : A F1, B F2}) (12,F1F2) (1,F1), (2,F2).

5. E 1 2 : E F1 F2. x 1 Ex = {y 2 : (x, y) E}. y 2, Ey = {x 1 : (x, y) E}. Ex F2, Ey F1 x, y.(Updeixh:Jewreste thn klsh: A = {A 1 2 : Ax F2 x}.Dexte ti h Aenai s-lgebra kai ti A C .)

0 : C 7 [0,] :0(AB) = 1(A)2(B)

pi 0 = 0 = 0, =pi 0 pi pipi 1.1.25

15

pi (),(),() pi pi pi - F1 F2. pi -pipi 1, 2 . :

1.4.1. (1,F1, 1) (2,F2, 2) pi 1, 2 -pipi. pi (12,F1F2) pi :

(AB) = 1(A) 2(B) A F1, B F2.

-pipi 1, 2. = 1 2.

1.4.2. (1 2,F1 F2, 1 2) pi (i,Fi, i), i = 1, 2 pi. pi pi :

E 1 E / F1 2 6= , F2 2() = 0. E 1 (12)(1) =1(1) 2() = 0 pi pi pi E F1 F2 ( . 6). y (E )y = E F1 pi.

6. (1, F1, 1), (2, F2, 2) pi (i,Fi, i) (1 2,F1 F2, 1 2) pi 1 2 :

i. F1 F2 F1 F2 F1 F2.

ii. (1 2)(E) = (1 2)(A) A F1 F2

iii. F1 F2 = F1 F2.

pipi Lebesgue Rm.pi = (Rm, Bm) = (Rm,M) Lebesgue Rm( . 12). pi m Lebesgue Rm.

1.4.3. k, ` N k+ ` = m Rm = RkR` :

1. Bk B` = Bm

2. Bk B` Bm ( pi k, `, m ).

3. (Bk B`) = Bm ( pi k `).

4. m = k `.

16

Gia ta mtra ginmena kai to mtro Lebesgue sumbouleutete to [12]

pi pi pi pi pi pi. pi - transition probability measurable kernel.

1.4.4. (X,A ) (Y,B). pi- ( pi) K : X B 7 R pi pi :

1. x X K(x, ) pi (Y,B).2. B B K(, B) A - pi X R

1.4.5.

1. (X,A ) = (Y,B) = (R,B1) > 0

K(x,A) =

A

12pi2

e(xy)222 dy, x R, A B1.

2. (X,A ) pipi (Y,B, Q) pi

K(x,A) = Q(A), x X, A B.

3. pi pi, Q

K(x,A) = x(A), x X, A B.

1.4.6. (X,A , P1) pi (Y,B) . K(x,B), (x,B) X B pi . pi pi P (X Y,A B) pi :

P (AB) =A

K(x,B)dP1(x) A A , B B.

( pi E A B : P (E) = X

K(x,Ex)dP1(x)).

1.4.7. P2 pi (Y,B) piK(x,) =P2() x X, B P

P (AB) =A

P2(B)dP1(x) = P1(A) P2(B)

P = P1 P2 pi ( pi).

17

Keflaio 2

Mtra se Topologikoc

Qrouc Hausdorff

2.1 Topologiko qroi

2.1.1. pi (X, T ) pi X 6= T pi X pi pi pi:

i. , X T

ii. Vi T i = 1, 2, ...n ni=1

Vi T

iii. Vi T i I iI

Vi T ( I).

pi X pi T ( piT ). pi T E T pi pi pi pi:

U T pi pi pi E ( pi

i

Ei = )

E pi T x X Nx = {U E :x U} Nx pi x pi T . pi x X A X pi pi U T x U A pi x X Nxpi pi x pi A x pi U Nx U A. 2.1.2. A T x A pi U Nx U A. pi pi pi pi pi.

19

2.1.3. X 6= E pi X pi pi pi pi:

1.{A : A E } = X

2. A,B E x A B pi E x A B. pi pi T X pi A T - pi pi E pi pi piT . pi pi x X pi Nx pi E =xXNx pi pipi pipi .

2.1.4. X pi d, d : X X 7 [0,) :

1. d(x, y) = d(y, x)

2. d(x, y) = 0 x = y3. d(x, y) d(x, z) + d(z, y) x, y, z.

x X, r > 0 B(x, r) = {y X : d(x, y) < r} E = {B(x, r) : x X, r > 0}. E pi pi pipi pi pi pi T pi E . pi A T x Api r > 0 : B(x, r) A.( r pi pi ).

2.1.5.

1. pi pi- T pi pi pi .pi E ,E pi X pi pi pi pi pi pi pi.. pipi (X, d) E = {B(x, r) :x X, r > 0} E = {B(x, r) : x X, r Q+}. pipi pi T , T pi pipi E ,E T = T .

2. pi T pipi : {xn} X x X. xn Tx U T x U pi n0 : n n0 xn U .pi pipi pi . T1

20

T2 pi xn T1 x xn T2 x. pi.. (1+ 1n )

n, n N pi (R, T2 = P(R)). pi (R, T1) T1 pi pi d(x, y) = |xy| e.

2.1.6. pi (X, T ). A X - Ac ( T ). X G .

2.1.7. pi (X, T ) :

1. n N A1, ...An ni=1

Ai

2. I Ai, i I iI

Ai

.

2.1.8. pi (X, T ) Hausdorff pipi x, y X x 6= y pi U, V T x U, y V U V = 2.1.9. (X, d) pi pi pipi d Hausdorff.

7. (X, T ) Hausdorff {xn} X x X. xn T x xn

T y. x = y.

pi pi Hausdorff. {x} pipi

.

2.1.10. (X, T ) pi Hausdorff . K X pi K (. Ui T , i I

iIUi K) pi pipi pi K (. pi

Ui1 , ...Uin nk=1

Uik K). pi pi X K .

2.1.11.

1. pi pi K X .2. F K pi F K pi F

pi.

3. K1, ...Kn K ni=1

Ki K .

21

4. Ki K , i I iI

Ki K .

2.1.12. pi pi Rm pipi pi d(x, y) = |x y| :

K Rm pi K . 2.1.13. pi (X, T ) A X. Ao =

{U T : U A} A = {F : F A}. Ao A . A .

Ao A A A = A \Ao .

2.2 Kanonik mtra se topologikoc qrouc

2.2.1. (X, T ) pi . - Borel pi pi X, B = (T ). 8. E pi. (X, T ) B = (E ).

pi - BorelB pi (X, T ) pi , pi, T ,G ,K B. 2.2.2. (X, T ) pi - A B piB - Borel. Borel X (X,B). (X,A ) pipi pi:

1. (K) < + pi K K2. A A :

(A) = inf{(U) : U U A}

3. U T :

(U) = sup{(K) : K pi K U}

: (X,A ) pipi K pipi A .

2.2.3. pi. (X, T ) - A B. (X,A ). pi:

22

1. A A A =n=1

An pi An A (An) < + ( A -pipi )

(A) = sup{(K) : K pi A} ()

() A A X =n=1

En En A (En) < + (. -pipi).

2. A A pi pipi :(A) = sup{(F ) : F A}

pi.

1. ( [3] . 208)

2. K G .

2.2.4. pi. (X, T ) (X,B) pipi pi:

1. (A) = sup{(K) : K pi A} A B.

2. X =n=1

Un Un (Un) < +.

. pi 2.pi pi (X) < +.

pi. pi K K K n=1

Un pi

K mk=1

Unk (K) mk=1

(Unk) < +. (B) = sup{(F ) : F B} B B K G . A B pi Uk (Uk \ A) = sup{(F ) : F Uk \A} pi > 0 pi pi F Uk \A

(Uk) (A) < (F ) (Uk) (A) (F ) = (Uk) (Uk \ F ) V = Uk \ F

(A) (V ) < (A) + pi V A

pi 2. pi X =n=1

n n B,

n Un, n N. ( pi 1 = U1 n = Un \ (n1i=1

Ui), n >

23

1).pi B B pi B =n=1

(B n) =n=1

Bn

pi Bn = B n, n N Bn Un n N. pipi Bn > 0 pi Vn Bn pi :

(Vn) < (Bn) +

2n, n N

U =n=1

Vn U B

(B) (U) n=1

(Vn) 0 uprqei sumpagcK f1(B) me (f1(B)\K) 0 pi pi K A (A) < (K) + pi = (A) (A)

(A) < (K) K A.pi :

Kc Ac

pi (Ac) (Kc). pi (X) < (X) - pi.

2.3 Mtra se topik sumpagec topologikoc

qrouc

2.3.1. pi. (X, T ) pi pi x X pi pi x pi x X pi U T x U U pi. 2.3.2. X = Rn T pi pi pi d(x, y) = |xy|. x Rm r > 0 B(x, r) = {y Rm : |yx| 0} pi T . pi pi pi. (X, T ) pi pi.pi pi pipi (X,B) (K) < + pi K Rm (pi pi Lebesgue.)

Gia apodexeic dec [6] [12]

pi pi Riesz Representation pi pi pipi. pi [12] pi. (X, T ) pi pi (Y, T ) pi. . f : X 7 Y pi T , T f1(U) T U T . (Y, T ) = (R, T ) T pi R pi pi . f : X 7 R x X : > 0 pi pi U x f(U) (f(x0) , f(x0) + ). f : X 7 R C(X,R) C(X). f C(X,R)

s(f) = {x X : f(x) 6= 0} f C(X,R) pi s(f) pi pi pi pi Kf K f(x) = 0 x X \Kf .K(X) pi , :

K(X) = {f C(X,R) : s(f) pi}= {f C(X) : K K f(x) = 0 x X \K}

K(X) pi ( pi f, g K(X) f(x) + g(x) = 0 x X \Kf Kg Kf Kg pi). I : K(X) 7 R I(f) 0 f K(X) f 0.

26

2.3.5. (Riesz Representation) (X, T ) pi pi I : K(X) 7 R. pi Borel (X,B) pi I(f) =

fd f K(X).

2.3.6. X = R T pi. I : K(R) 7R : I(f) =

Rf(x)dx pi Riemann

( pi pi [a, b] Kf ). pi- ,: pi (R,B1) pi I(f) =

Rf(x)d(x) f K(R). -

pi Lebesgue (R,B).

pi pi pi pi pi pi - norm pi.:

2.3.7. X pi. . pi pi pipi .

pi. [21] . 17.

2.4 Kataskeu kanonikn mtrwn

pi pi pi pi. Hausdorff. pi pi - .

2.4.1. pi. (X, T ) (Hausdorff) : B 7 [0,) pi pi pi pi:

i. A,B B A B = (A B) = (A) + (B)

ii. A B (A) = sup{(K) : K pi A}

(X,B).

pi. {Bn} B Bn Bn+1 n=1

Bn = limn(Bn) = 0

(ii.) > 0 n N pi piKn K Kn Bn (Bn Kn) 2n+1 .

i=1

Ki =

27

pi ( pi) pi n0 N n0i=1

Ki = .pi :

Bn0 = Bn0 \n0i=1

Ki =

n0i=1

Bi \n0i=1

Ki n0i=1

(Bi \Ki)

(Bn0) n0i=1

(Bi \Ki) n0i=1

2i+1< .

{Bn} (Bn) < n n0.pi 2 . 10 pi -pi, {An} B : (

nAn) =

n(An). pi

2.2.4.

2.4.2. pi. Hausdorff : K 7 [0,) pi pi pi pi:

i. K1 K2 (K1) (K2)ii. (K1 K2) (K1) + (K2)

iii. (K1 K2) = (K1) + (K2) K1 K2 = iv. > 0 K K pi U K (C) < (K) +

C K K C U . pi pi (X,B).pipi sup{(K) : K K } = 1 pi.

pi. pi !

2.4.3. (iv.) pi : (iv.) > 0 K K pi U T U K (C) (K) + C K : C U . (iv.) pi () < (K) + K : K U C K C U K C K K K C U pi (K C) < (K) + (C) < (K) + pi - . pi pi pi - [16].

2.4.4. (X, T ) pi. Hausdorff A pi X A BX . : A 7 [0,) pipi

i. (X) = 1

28

ii. A,B A AB = (AB) = (A)+ (B) (pi pi)

iii. B A (B) = sup{(F ) : F , F A , F B}

iv. > 0 pi pi K X pi (B) > 1 B A B K

v. F pi pi T .

pi (pi) (X,BX) pi pi- .

pi. pi

pi i pi (X) = a > 0 iv (B) > 1 pi (B) > a . pipi pi. pi Henry pi pi .

2.4.5. (X, T ) pi. Hausdorff A pi X A B. : A 7 [0,) pipi:

() A,B A AB = (AB) = (A)+(B) (pi pi).

() A A (A) = sup{(K) : K pi A K A }.

pi (X,B). pipi Api pi T (X,B) pi (A) = (A) A A .

pi Henry pi [15] [20] pi - pi pi pipi pi A pi pi T ( pi ). Zorn pipi . pi pi Henry , , (X) < + , pi () (a) () ii, iii, iv 2.4.4. pi pi. ( (B) > (X) (B) > 1 iv). . pi .

2.4.6. pi Henry (), () pipi ()

() (A) = sup{(F ) : F A F A }

() > 0 pi pi K A pi (K) > (X) .

29

pi. pi (a) pipi pi pipi. pi > 0 A A pi (), ()

(A \ F) < 2

F A F A

(X \K) < 2

K pi A

A \ F K = X A \ F K (X \K) (A \ F) pi = F K (A \ ) < . = F K A pi.

pi pi pi pipi. . pi. pi. pi pi pi (com-pletely regular ).

2.4.7. pi. (X, T ) pi (com-pletely regular) Hausdorff x X F X x / F pi f : X 7 [0, 1] f(x) = 1 f(y) = 0 y F . 2.4.8. (X, d) pi . f(y) = d(y,A)d(x,A) , y X.

2.4.9. (Prohorov) (X, T ) pi (completely regular ) pi. C(X) pi . C(X) pi . pipi x 6= y X pi f f(x) 6= f(y). A pi X {x X : (f1(x), ..., fn(x)) B} pi {f1, ..., fn} B Bn- - Borel Rn. A pi X () A B.pi : A 7 [0, 1] pi :

i. (X) = 1

ii. A,B A A B = (A B) = (A) + (B)iii. A A (A) = sup{(F ) : F A F A }iv. > 0 pi pi K pi : (B) > 1

B A B K. pi pi (X,B) pi pi .

pi. pi pi pi pi pi .

30

2.4.10. pi. Hausdorff X - RX . pi (X,()) A ()

(A) = sup{(F ) : F X ,F ()}

2.4.11. pi pi. X - BaireB0(X) = (C(X,R)). B0(X) pi pi TX X.

2.4.12. Y pi. Hausdorff -pi, pi

pi Kn, n N Y =n=1

Kn C(Y,R) pi Y . - Baire B0(Y ) (C(Y,R)) B0(Y ) = ().

, pi pi A a(X,). A = a(X,). pi

() = (a(X,)) = (A )

pi pipi (pi ii, iii) - :

() = inf{(U) : U , U A } , A . (1)

pipi pi ( 11 pi) - -pi. o pi pi (,A ). o (pi) (X,(A )) pi

o|A = pi: K pi X (B) > B A B K o(K) pi (0, 1). , > 0 pi pipi pi n A

n=1

n K

n=1

(n) < o(K) + (2)

pi pi (1) pipi Un A Un n (Un) < (n) +

2n pi

n=1

(Un) (4)

pi

(V ) = o(V ) n=1

(Un) (5)

pi (2),(4),(5) pipi

(V ) < o(K) + +

pi pi (4)

o(K) + + > , > 0

pi pi . pi pi iv pipi n N pi piKn X (B) > 1 1n B A B Kn. pi o(Kn) > 1 1n , n N pi Y =

n=1

Kn

o(Y ) = 1

pi pi pi 1.1.29. pi o

pi (Y, (A )Y ) pi (A )Y = { Y : (A )} ( Y ) = o(). pi 1.1.28.

(A )Y = (AY ) pi AY = {A Y : A A }

Y pi pi TY = {U Y :U TX} Y = {f |Y : f } C(Y,R). AY = a(Y,Y ) pi

(A )Y = (Y )

pi Y -compact Y Y , 2.4.12.

(Y ) = B0(Y ) - Baire Y

pi pi (Y,B0(Y ), ) pi pi:

. (A) = sup{(F ) : F X, F B0(Y )} pipi pi B0(Y ) 2.4.10. = C(Y,R).

32

. > 0 pi pi K Y (A) > 1 A B0(Y ) A K. pi pipi m N 1m < o(Km) > 1.pi A B0(Y ) A K Km (A) = (A Y ) =o(A) o(K) > 1

. B0(Y ) pi pi TY . X pi pi 2.2.11. - B0(X)pi E pi TX . E B0(X) piEY = {U Y : U E } B0(X)Y . EY piTY pi B0(X)Y B0(Y )

pi (),(),() pipi pi pi 2.4.4. (Y.B0(Y ), ) pi pi pi pi (Y,B(Y )). pi B(Y ) = B(X)Y : B(X) 7 [0, 1] (B) = (B Y ). pi pi (X,B(X)) pi A A AY B0(Y )

(A) = (A Y ) = (A Y ) = o(A) = (A)pi pi. pi 1,2 (X,B(X)) 1 = 2 A . n N pi pi Kn X 1(Kn) >1 1n 2(Kn) > 1 1n . Y =

n=1

Kn. 1(Y ) = 2(Y ) = 1.

1.1.29. pi i (i = 1, 2) B(X)Y

i(B Y ) = i(B)

1 = 2 AY pi

1 = 2 (AY ) (6)

Y pi pi TY pi B(X)Y = B(Y ) 1, 2 pi (Y,B(Y )). pipi, pi pi pipi (AY ) = B0(Y ) : - Baire B0(Y ) pi pi TY . pi- 14 (6) pi

1 = 2 B(Y )

pi 1 = 2 B(X).

pi .

33

pi. 2.4.10. A (). pi 37,27 26 pipi 3 4 - pi pi pi - (1, 2, ...) pi pi

A = 1(B) pi = (1, 2, ...) : X 7 RN B B(RN).

() B(RN) TX pi RN. (RN,B(RN)) pi

(B) = (1(B))

( ;) pi > 0 pi E RN pi

(B \ E) <

(1(B) \ 1(E)) < 1(B) = A 1(E) X.pi. 2.4.11. z X U pi z. z / U c pipi f C(X,R) f(z) = 1 f(x) = 0 x U c. V = {x X : f(x) > 0} :

V z V U

pi {x X : f(x) >0} f C(X,R). -Baire B0(X) = (C(X,R)).

pi. 2.4.12.pi C(Y,R) () B0(Y ). pi Y () B1 . Y . pi - pi pi C(Y,R) pi . C(Y,R) pi pi Stone-Weirstrass pi C(Y,R) f C(Y,R) -pi {fn, n N} pi K Y

supxK|fn(x) f(x)| 0

pi Y =n=1

Kn Kn Y pi y Y y K` pi ` N fn(y) f(y). f C(Y,R)

34

f = limnfn fn pi f C(Y,R) ()B1 .

pi

B0(Y ) ().

pi pi pi pi . X pi. pi. . = X . ( 2.4.1. pi ) pi :

2.4.13. pi. (X, T ) Hausdorff (pipi) pipi (X,B). pi :

i. (A) = sup{(F ) : F A} A Bii. > 0 pi K pi (K) > (X) (tightness)

( 2.2.2.) A B:

(A) = sup{(K) : K pi A}pi. pi pi 2.4.6. pi pi 2.2.4.

pipi pi pi pi pi pi.X pi. A B.

1. (A) = inf{(U) : U A}2. (A) = sup{(F ) : F A}3. (A) = sup{(K) : K pi A}4. > 0 pi K (K) > (X) pipi X Hausdorff

(3)(2) + (4)m(1)

pipi 2.2.4.

10. pi. (X, T ) (X,B). (X,B) .(Updeixh: Prpei na deiqte mno ti (B) = inf{(U) : U anoikt B} gia tuqnB = A N me A B kai N N.

35

An (A) = + -ekolo. An (A) < + tte gia tuqn > 0 uprqei anoikt U A me (A) (U) 0.Afo

n

Fn = ja uprqei(ap 2) na peperasmno J N me (

iJFi) 0 f Lb(X) - - pi . W pi pi

aW liminf fda fd

f Lb(X).

pi pi [15]

53

Keflaio 3

Probolik sustmata

mtrwn

Mtra se kartesian

ginmena aperwn

paragntwn

3.1 Topologik probolik sustmata mtr-

wn

pi pi pi pi pi:

. I pipi: pipi i, j I pi ` I i, j `.

. pi Hausdorff (Xi, Ti), i I pipi i (Xi,Bi) pi Bi = (Ti) - Borel Xi, i I.

. pipi i j pi Pij : Xj 7 Xi pipi pi :

i. Pii : Xi 7 Xi i I.ii. Pi` = Pij Pj,` i j ` Iiii. j(P

1ij (B)) = i(B) i j B Bi

(Xi, i, Pij), i I.

55

3.1.1.

pi (R,B1). I = {i N : i pipi} . Xi = R|i|, i I pi

i =|i|1, i I (R|i|,B|i|)

Pij = piij : R|j| 7 R|i| pi.

3.2 Jerhma Prohorov

pi R pi pi ( pi) i, i I. pi - I -pipi. pi pi pi:

3.2.1. (Prohorov) pi pi (Xi, i, Pij), i I i(Xi) 0 pi pi K X

i(Xi \ Pi(K)) < i I (P1i (B)) = i(B) i I B Bi. pi pi ( pi (X) = i(Xi) i I).

pi. pi !

Efarmogc

56

I. T pipi {St, t T} pi. Hausdorff. I = {i T : i pipi} pi. X =

tTSt =

{x : T 7 tT

Xt x(t) St t T} i I,Xi =tiSt.

i I, pii : X 7 Xi i-pi, pii(x) = x|i. pi pii pi. i = {t} pit = pi{t}, t T . Tt pi St, t T pi X :

{tipi1t (Ut) : i I, Ut Tt t i}

pi i I pi Xi =tiSt {

tiUt :

Ut Tt t i}. pi pi ( pi) pi Tt, t T . i j I piij : Xj 7 Xi pi piij(y) = y|i pipi pi X Xi pi pii, i I piij , i j , . {pii, i I} X. pi. Xi, X - Borel Bi B, i I. pi :

tiB(St) Bi, i I

tTB(St) B

pi tB(St) (pit, t ) = ({pi1t (B) : t , B B(St)})

3.2.2. (Kakutani)pi pi. {St, t T} pi pipi-, i (Xi,Bi), i I pi :

i(B) = j(pi1ij (B)) i j I B Bi.

pi (X,B) : (pi1i (B)) =i(B) i I B Bi. (X) = i(Xi) i I.

pi. (Xi, i, piij), i I pi pi pi . pipi pi. X pi pi Prohorov . > 0 pi pi K = X i(Xi \pii(K)) = i(Xi \Xi) = 0 < i I. ( pii pi).

57

II. {Sk, k N} pi. Hausdorff pi. (pi Hausdorff)

Xi =

ik=1

Sk, i = 1, 2, ...

X =

k=1

Sk

pi:

pii,i+1 : Xi+1 7 Xi

pii : X 7 Xi, i N

pi pi , ( pi) pi .pi pipi i (Xi,Bi), i I pi pi :

i(B) = i+1(pi1i,i+1(B)) i N, B Bi.

pi :

i(B) = j(pi1i,j (B)) i j, B Bi

pi pii,j : Xj 7 Xi pi. pi pi pi (Xi, i, pii,j), i N pi {pii, i N} - X.

3.2.3. {Sk, k N} pi. Hausdorff pipi- i (Xi,Bi), i N pi Xi =i

k=1

Sk. pi i, i I pi :

i(B) = i+1(pi1i,i+1(B)) i N, B Bi ()

pi (X,BX)

pi X =k=1

Sk pi pi :

(pi1i (B)) = i(B) i N, B Bi

: (X) = i(Xi) i N.

58

pi. Prohorov., > 0 pi pi Li Xi, i N . 1 piL1 X1 :

(X1 \ L1) < 22

(1)

2 pi L2 pi11,2(L1) pipi :

2(pi11,2(L1) \ L2) = 0 x E} 6= kai ra uprqei x0 X \ {0} me< x, x0 >= 0 gia kje x

E. An tra : X 7 X enai h kanonik apeiknish(x)(x) =< x, x > lgw anaklastikthtac tou X ja enai (X) = X kai ra up-rqei x0 X \ {0} me (x0) = x0 kai ra < x0, x >= 0 gia kje x E - topoafo x0 6= 0 kai h G diaqwrzousa.

40. X,Y pi pi. . T : X 7 Y . C(X,X ) C(Y, Y ) .Updeixh: y T X gia kje y Y ra C(X,X )B1 metrsimh. Epikalestetetra thn 'Askhsh 36.

pipi pi pi -C(X,) X, . pi.. X pi pi. . = X . pi pi f = (f1, ..., fk) {(f1, ..., fk) : k N, fi } :

Af = {f1(B) : B Bk}

pi - C(X,) pi pi f = (f1, ..., fk) f1, ..., fk .

4.2.9. X RX . . L = {(f1, ..., fk) : k N, fi }. pi:

1. C(X,) = (fL

Af )

2. f = (f1, ..., fn) L g = (g1, ..., gr) L pi h =(h1, ..., hm) L Af Ag Ah. pi 1 :Rm 7 Rn 2 : Rm 7 Rr pi :

f = 1 h g = 2 h

83

3.fL

Af =f

Af = C(X,).

pi.

1. pi {1(B) : \ {0}, B B1} fL

Af f

Af

pi 35 C1(X,\{0}) (

fLAf ) (

f

Af ). (f

Af ) = C(X,)

pi C1(X, \ {0}) = C1(X,) = C(X,).2. pi pi pi {f1, ..., fn, g1, ..., gr}

{h1, ..., hm} E.

fk =

m`=1

ak`h` , k = 1, ...n

h = (h1, ..., hm) : Rm 7 Rn pi (u1, ..., um) =[ak` ](u1, ..., um)

> h L f = hpi B Bn f1(B) = h1(1(B)) 1(B) Bm Af Ah. g.

3. F . f = (f1, ..., fn)

fk =

mk`=1

bk` gk` , k = 1, ..., n

pi {gk1 , ..., gkmk} F (k = 1, ..., n). pi pi pi {gk` : ` = 1, ...,mk, k = 1, ..., n} fk E, k = 1, ..., n pi {h1, ..., hm} E

fk =

m`=1

ak`h` , k = 1, ..., n

h = (h1, ..., hm) : Rm 7 Rn pi (u1, ..., um) =[ak` ](u1, ..., um)

> f = h Af Ah h L.

4.2.10. pipi = X pi X pi pi. . X pi , X =(X, ). pi :

C(X,X ) B(X,(X,X )) B(X)

pi B(X) = B(X, ).

84

4.3 Kulindrikc s-lgebrec ston duk qro

- X = (X, ) pi pi. . pi . :I = {i X : i pipi} i = {x1, ..., xn} IAi = {g1i (B) : B Bn} pi gi : X 7 Rn pi gi(x) = (, ..., < xn, x >). C(X , X) =iIAi C(X , X) = (

iIAi).

C(X , X) - X pi - () C(X , X). pi X pi (X) X : X 7 X pi pi

(x)(`) = `(x) , ` X

pipi

C(X , X) C(X , X ) B(X ,C (X , X))

pi C (X , X) pi X . C(X , X ) pi pi pi - X = X . E. Mourier X pi

4.3.1. X norm X C(X , X) = B(X ). ( pi X pi x =sup{| < x, x > | : x 1}).

4.4 Mtra kai kulindrik mtra pijanthtac

se t.d.q. Jerhma Prohorov

pi pi pi . pi pi pi pi pi pi . pi pi ( pi) - - Borel pi pi pi . pi.. X pi ... pi C (X,X )- pi - Borel B(X,C (X,X )) . pi - Borel B(X, ) B(X). pipi pi :

4.4.1. pi 1, 2 X T1, T2. pi T1 T2 ( 2 pi 1). 2- pipi 1-.

85

pi. B1 (T1) (T2) B2 pi B1. K1,K2 pi pi 1, 2 pi K2 K1 pi A B1

(A) = sup{(K) : K K2,K A} sup{(K) : K K1,K A} (A)

pi 2.2.4.

4.4.2. pipi pi pi

pi -pipi . X =n=1

Un Un 1-

(Un)

4.4.5. X RX pi ( X- X). {f1, ..., fn} f = (f1, ..., fn). pi f : X 7 Rn pi.pi. y = (y1, ..., yn) Rn \ {0}. (Robertson& Robertson: Top. Vector Spaces, . 33) pi a1, ..., an X pi fi(ai) = 1 fi(aj) = 0 j 6= i (i = 1, ..., n). x = y1a1 + ...+ ynan. fi(x) =

nj=1

yjfj(aj) = yi f(x) = y.

4.4.6. . X . - X (X X). pi f = (f1, ..., fn) L pi f (Rn,Bn) pi pi pi :

f = (f1, ..., fn) L h = (h1, ..., hm) L f = hpi : Rm 7 Rn :

h(1(B)) = f (B) , B Bn ()

pi :fL

Af 7 [0, 1] (X) = 1 -pi pi - Af , f L pi :

(f1(B)) = f (B) f = (f1, ..., fn) L , B Bn ()

pipi -pi - Af , f pi C(X,).

pi.

: C(X,) fL

Af 7 [0, 1] (A) = f (B) A = f1(B) Af . A = f1(B1) = g1(B2) pi f = (f1, ..., fn) L g = (g1, ..., gr) L,B1 Bn B2 Br. 4.2.9. pi h =(h1, ..., hm) L f = 1h g = 2h pi 1 : Rm 7 Rn 2 : Rm 7 Rr pi:

A = h1(11 (B1)) = h1(12 (B2))

h1(11 (B1) \ 12 (B2)) = pi pi h pi

11 (B1) 12 (B2)

87

pi pi

11 (B1) = 12 (B2) (1)

pi () :

f (B1) = h(11 (B1)) g(B2) = h(

12 (B2))

pi (1) pi f (B1) = g(B2). -pi - Af f L. {An, n N} Af An = f1(Bn), Bn Bk f1(Bi Bj) = f pi Bi Bj = i 6= j. pi:

(n

An) = (f1(n

Bn)) = f (n

Bn)

=n

f (Bn) =n

(An)

pi (X) = 1 pipi .pi . f = (f1, ..., fn) . fi 6= 0 pi pi pi {f1, ..., fn} {h1, ..., hm} . h = (h1, ..., hm) L f = h pi :Rm 7 Rn pi B Bn f1(B) = h1(1(B)). pi pi Af Ah -pi Af f . fi = 0 i = 1, ..., n Af = {, X} . pi C(X,) =

fL

Af =f

Af ( 4.2.9.)

pi .

4.4.7. pipi pi pipi 4.4.3.

4.4.8. . X X (pi X X). pi f = (f1, ..., fn) pi f (Rn,Bn) pi pi :

f = (f1, ..., fn) h = (h1, ..., hm) f = hpi : Rm 7 Rn :

h(1(B)) = f (B) , B Bn ()

pi pi : C(X,) =f

Af 7[0, 1] pi :

(f1(B)) = f (B) f = (f1, ..., fn) B Bn ()

88

pi. L pi f , f L pi C(X,) pi pi ().pi pi (A). f = (f1, ..., fn) fi 6= 0 B Bn pi f = h pi h L pi (f1(B)) = (h1(1(B))) = h(1(B)). pi () (f1(B)) = f (B). pi f = (0, ..., 0) pi () f = 0. pi (f1(B)) = 0.

4.4.9. pi pi pi pipi:

f , f L pi . -, pi . pi Pro-horov pi . pi:

4.4.10. X pi pi. . . pi C(X,R). pi X. A C(X,)

(A) = sup{(F ) : F C(X,), F A}( pi X C(X,R)).

pi. A C(X,). pi f L A Af A =f1(B) f = (f1, ..., fn) L B Bn. pi(C) = (f1(C)), C Bn (Rn,Bn). pi > 0 pi E Rn E B (B \ E) < . f F = f1(E) pi C(X,) ( ) pi f1(B) = A. (A \ F ) = (f1(B \ E)) = (B \ E) < .

4.4.11. (Prohorov ) X pi pi. . . C(X,R) pi X. pi X pi pi :

> 0 pi pi K X pi (*)

(A) > 1 A C(X,) A K pi pi (X,B(X)).

pi. pi pi:

X pi (completely regular )

89

C(X,) = f

Af pi = {(f1, ..., fk) : k N, fi }

pi pi (A) = sup{(F ) : F A,F C(X,)} Prohorov (*)

pi 2.4.8.

41. Prohorov (*) pipi: > 0 pi pi K X pi

vf (f(K)) > 1 f = (f1, ..., fn) L

pi vf pi (Rn,Bn) pi pi

vf (B) = (f1(B)) , B Bn.

Updeixh: 'Estw ti isqei h (*). 'Estw f = (f1, ..., fn) L. Lgw sunqeiacf(K) Bn kai ra f1(f(K)) C(X,). 'Omwc f1(f(K)) K kai ravf (f(K)) = (f

1(f(K))) > 1 .Antstrofa: 'Estw A C(X,) me A K. 'Omwc A = f1(B) gia kpoio f =(f1, ..., fn) L kai ra f(f1(B)) f(K). 'Omwc B f(f1(B)) kai ra vf (B) vf (f(K)) > 1 .'Omwc vf (B) = (f1(B)) = (A).

pi pi pipi Haar ( ) pi.

4.4.12. (A. Weil) X pi. . Hausdorff. pi (X,B) pi pi (. (a + B) = (B) a X B B) pi. X pi pi.pi. [16] .73

4.4.13. X pi. Hausdorff pi. pi (X,B) pi -.

pi. pi X pi pi pipi - pi.

90

Keflaio 5

Qarakthristik

sunarthsoeid mtrwn

pijanthtac

5.1 Orismo kai idithtec twn qarakthris-

tikn sunarthsoeidn

pi Rn pi

(t) =

Rnei(x,t)d(x) , t Rn (1)

: Rn 7 C Rn:

pipi t1, ..., tm Rn pipi c1, ..., cm C:

mk,`=1

ck c`(tk t`) 0

pipi (0) = 1 : Rn 7 C Rn. pi .

5.1.1. (Bochner) pi pi :

i : Rn 7 C , (0) = 1ii pi pi (Rn,Bn)

.

91

pi. [9]

pi pi.. pipi .

5.1.2. Y pi (pi.. . ). X : Y 7 C pi-pi m N, t1, ..., tm Y c1, ..., cm C :

mk,`=1

ck c`X (tk t`) 0

X : Y 7 C pi pipi pi :

1. X (t) = X (t) t Y2. |X (t)| X (0) t Y3. |X (t1)X (t2)|2 2X (0)[X (0)ReX (t1t2)] pipi t1, t2 Y4. Xn, n N eX 5. X1,X2 pi Y1, Y2

X (t1, t2) = X1(t1) X2(t2), (t1, t2) Y1Y2 Y1 Y2.

42. X : Y 7 C pi - Y ReX y = 0 X Y .

Updeixh: Epikalestete thn 3.

pi (1) t X pi X Hilbert (, ), pi - pi x 7 (x, t) . pi Hilbert ` pi `(x) = (x, t(`))pi (1) t X

X (`) (t(`)) =X

ei(x,t(`))d(x) =

X

ei`(x)d(x)

pi pi - pi pi pi. . .

5.1.3. pi . pi. X . RX . pi (X, C(X,)).

92

(..) X : 7 Cpi pi

X (f) =X

eif(x)d(x) , f

.. X .pi pi pi pi (C). (R,B1) vf , f vf (B) =(f1(B)), B B1 X (f) =

Reizdvf (z), f pi

(f) = vf (1), f (2)pi vf .. vf R.

5.1.4. ..

1. .. (0) = 1

2. {f, fn, n N} limnfn(x) = f(x) x X lim

n(fn) =

(f).

pi.

1. g = (f1, ..., fn) n (Rn,Bn) vg(B) =(g1(B)) , B Bn. y Rn y g =y1f1 + ...+ ynfn

(y g) =X

ei(yg)(x)d(x) =Rneiyzdvg(z)

(y g) = vg(y) , y Rn (3)

pi vg .. vg Rn.pi k, ` {1, ..., n}

(fk f`) = vg(zk z`)pi zk Rn 0 k pi 1. pi Bochner .. vg.

2. pi Lebesgue .

5.1.5. pi ... X pi X .

93

1. .. pi (X , X) pipi .

2. X norm .. pi C (X , X) pi pi ( pi pi norm x = sup{| < x, x > | : x 1}).

pi.

1. pi (2) pi pi

2. (X , X) C (X , X) . C (X , X) pi pi norm x - pi C (X , X) Re . .. pi (3) pi- (pi pi C (X , X) ).

pi pi - pi pi pi :

5.1.6. pi 1, 2 (X, C(X,))pi pi RX 1 = 2 . 1 = 2 C(X,).

pi. 1 = 2 C(X,) pi pi C(X,). C(X,) =

f

Af pi = {(f1, ..., fk) : k N, fi } Af = {f1(B) : B Bk} f = (f1, ..., fk). pi f 1 = 2 Af . , f = (f1, ..., fn) vi(i = 1, 2) (Rn,Bn) vi(B) = i(f1(B)). y Rn y f = y1f1 + ... + ynfn (pi pi (1) 5.1.4.)

i(y f) = vi(y) , y Rn (i = 1, 2) pi pi v1 = v2 ( Bochner) v1 = v2 Bn 1(f

1(B)) = 2(f1(B)) B Bn. 1 = 2 Af .

5.1.7. pi ... X pi C(X,R) pi X. pi- (X,B(X)). 1 = 2 1 = 2 B(X).

94

pi. pi pi 1 = 2 C(X,) 1 = 2 A = C(X,). pi X pi. pi. ., pi (completely regular ) pi 1, 2 pi pi pi 2.4.9.

5.1.8. 1, 2 pi (X, C(X,)) pi- f pi v1f , v2f (R,B1) pi pi : v1f (B) = 1(f

1(B)) v2f (B) = 2(f1(B)) , B B1 .

1 = 2.

pi. (2) 5.1.3. :

1(f) = v1f (1) 2(f) = v

2f (1) f

v1f , v2f .. v

1f , v

2f pi 1 = 2.

43. X = RN pi .

1.

RNo {(a1, a2, ...) : pi k N ai = 0i k}

2. C(X,X ) = B(X).

3. 1, 2 pi (X,B(X)) 1(y) = 2(y) y RNo . 1 = 2.

5.2 Qarakthristik sunarthsoeid kulindrikn

mtrwn

pi (X,C(X,)) pi pi pi pi -Af , f pi C(X,). .

5.2.1. pi (X,C(X,))pi pi RX . - X : 7 C pi pi X (f) =

X

eif(x)d(x) , f pi f (X, {f1(B) :B B1}, ). .. . .. pi pi .. pi pi pi pi:

95

5.2.2. pi RX . :

1. 1, 2 pi (X,C(X,)) 1 = 2 1 = 2.

2. pi (X,C(X,)) .. : 7 C (0) = 1 pipi pi .

pi. pi 5.1.6. (1) 5.1.4. {f1, ..., fk} pi norm pi pipi : g = y21 + ...+ y2k g = y1f1+...+ykfk =y f f = (f1, ..., fk).pi (Robertson & Robertson . 37) pi pi pi pi X pipi pi pipi pi norm pi pi gn = yn f y f = g pi |yny| 0pi | | norm Rk. pi (3) 5.1.4. pi .

pi pi pipi pi pi pi :

5.2.3. pi (Rm,Bm) :Rm 7 Rn > : Rn 7 Rm. pi (Rn,Bn) (B) = (1(B)) , B Bn. :

(z) = (>(z)) , z Rn (4)

pi , (Rn,Bn) (Rm,Bm) (4) (B) = (1(B)) B Bn.pi. [aij ] nm pi . z = (z1, ..., zn) Rn t = (t1, ..., tm) Rm z (t) =

ni=1

mj=1

ziaijtj =mj=1

tjni=1

aijzi =

t [aij ]>z z (t) = t >(z) (5)

,, Rm

eiz(t)d(t) =Rneizyd(y) = (z)

(5) pi Rm

eit>(z)d(t) = (>(z))

96

(4). pipi (B) = (1(B)), B Bn ..(z) = (>(z)), z Rn (4) = .

pi pi pi 4.4.6.

5.2.4. .. X pi X (pi X X). X : 7 C X (0) = 1 ( pipi pi ). pi pi X .. = X .

pi. L = {(f1, ...fn) : n N {f1, ..., fn} }. f = (f1, ..., fn) L f (y) = X (y f), y Rn. f (0) = 1 pi (1) 5.1.4. pi f Rn. Rn. V pi pi pi {f1, ..., fn} normg =

y21 + ...+ y

2n g = y1f1 + ...+ ynfn. yk y Rn

yk fyf = (yky)f = |yky| pi ykfyf 0. pi pi X pi V X (ykf) X (yf)pi f (yk) f (y). f pi pipi Bochner pipi pi f (Rn,Bn) pi f = f . h = (h1, ..., hm) L : Rm 7 Rn f = h pi y h = >(y) h pi pipi pi (5) :

X (y f) = X (y h) = X (>(y) h) pi

f (y) = h(>(y)) , y Rn

pi pi f h (Rn,Bn) (Rm,Bm)

f (B) = h(1(B)) , B Bn

pi 4.4.6. pi pi pi f = (f1, ..., fn) L B Bn

(f1(B)) = f (B) (6)

g (g) = eig(x)d(x) (6) (g) =eizdg(z) = g(1) = g(1) = X (g).

97

pi pi- .. . pi Hausdorff X C(X,R). pi B(X) C(X,) pi pi (X,B(X))

(f) =

X

eif(x)d(x) , f

P (X) pi (X,B(X)). P (X) pi W .

5.2.5. X pi ... pi C(X,R). {a, a I} P (X). :

1. aW P (X) a(f) (f) f .

2. pi X {a, a I} pi (P (X),W ) lim

aa(f) = X (f) f pi X : 7 C.

pi P (X) = X a W .pi.

1. eif(x) = cos f(x)+i sin f(x) .

2. {a, a I} (P (X),W ). 1, 2 P (X) {a, a I}. pipi {k}, {} k W 1 W 2. pilim k(f) = lim (f) = lim a(f) = X (f) f . pi (1) lim

ak(f) = 1(f), lim

a(f) = 2(f) f

pi 1 = 2 = X . 5.1.7. 1 = 2. pi {a, a I}. pi pi a

W . pi (1) pi = X .

5.2.6. pipi pi pi- .. pi pi pi. pi pi pi pi ( [16]).

98

Keflaio 6

Jewrmata Minlos kaiSazonov

pi 5.2.4. pi pi. pi (pi) . pi pi (pi) Minlos Sazonov pi pi -. pi [16]. - pi pi .

6.1 Summetriko telestc

6.1.1. X pi ... X . R : X 7 X x, y X :

< Rx, y >=< Ry, x > (1)

x X

< Rx, x > 0

Hilbert X ' X (-pi Riesz) (1) pi

(Rx, y) = (Ry, x) x, y, X

pi

(Rx, x) 0 x X

99

pi pi . ... X pi . pipi .

6.1.2. R : X 7 X ,. :1. (x, y) =< Rx, y > ,

2. Cauchy-Schwartz

| < Rx, y > |2 < Rx, x >< Ry, y > x, y X

pi.

1.

2. pi < R(x + y), x + y > 0 R.

6.1.3. R : X 7 X . :1.

2. (X , X) (X,X ) 3. (X , X) (X,X ) 4. C (X , X) C (X,X )

pi.

1. x, y X ` X `(R(x + y)) =< R(`), x + y >=< R(`), x > + < R(`), y >=< Rx, ` > + < Ry, ` >= `(Rx +Ry)

2. {xa} X xa x y X :< Rxa, y

>=< Ry, xa >< Ry, x >=< Rx, y > Rxa Rx pi (X,X ).

3. p, E q, E seminorm pi pi(X,X ) (X , X) . E ,pi (X , X) - pi = R() ,pi (1)(X,X )-pi, E .pipi y X p(Ry) = sup

x| < Ry, x > | =

supx| < Rx, y > | sup

x| < x, y > | pi

p(Ry) q(y) y X

100

4. norm pB , B D qA, A D pi C (X,X ) C (X , X) .

6.1.4. R : X 7 X , . : X 7 R pi (x) =< Rx, x > C (X , X)-.

pi. {xa} X xa C x X . pi C (X , X) (X,X )- A X

supuA|xa(u) x(u)| < a > a0(, A)

B = {xa} C (X , X)-pi pi (X , X)-pi = R(B) (X,X )- pi pi

| < Rxa, xa > < Rxa, x > | < a > a0

| < Rxa, xa > < Rx, x > | | < Rxa, xa > < Rxa, x > |+ | < Rxa, x > < Rx, x > |< + | < Rxa, x > < Rx, x > | a > a0

pi R pi pipi pi pi- .

6.1.5. pi pi X Banach pi C (X , X) pipi pi pipi pi norm X pi x = sup{| < x, x > | : x 1}.

6.1.6. R : X 7 X , . qR(x

) =< Rx, x >, x X seminorm X .

pi. Cauchy-Schwartz pi

q2R(x + y) =< Rx, x > + < Ry, y > +2 < Rx, y >

< Rx, x > + < Ry, y > +2< Rx, x >

< Ry, y >

= (qR(x) + qR(y))2

pi seminorm pi.

101

6.2 H topologa Sazonov

6.2.1. H Hilbert. T :H 7 H pi (nuclear) pi {xn, n N}, {yn, n N} H

nxn yn x X , y H). u : H 7 X pi norm H pi pi . X piX = (X, ).

6.2.2. pi ... X X .

1. R R(X , X) ,.2. qR(x) =

< Rx, x > seminorm.

3. seminorm qR, R R(X , X) X .pi.

1. < Rx, y >=< u(S(u(x))), y >=< S(u(x)), u(y) > pi u. S pi < S(u(y)), u(x) > pi < u(S(u(y))), x >=. pi pi pipi x = y S pi R.

2. pi 6.1.5.

3. x0 6= 0. pi x0 X \{0} x0(x0) 6= 0. pipi pi H X x0 H. pi pi H . (Robertson & Robertson. 37).

102

u : H 7 X u(x) = x. pi u x0(x0) 6= 0 pipi u(x0) a 6= 0. S0 : H 7 H S0y = (y, a)a. S0 S(H) pi R0 = u S0 u R(X , X).pipi R0x0 = u(S0(a)) = |a|2u(a) pi | | norm Hpi < R0x0, x

0 >= |a|2 < u(a), x0 >= |a|2(a, u(x0)) = |a|2 (a, a) =

|a|4 > 0.

6.2.3. X pi ... X . pi Sazonov X S(X , X) pi pi pipi seminorm {qR, R R} pi qr(x) =

< Rx, x >, x X .

... (X , S(X , X)) Hausdorff pi .

6.2.4. pi pipi pi 6.1.4.

S(X, X) C (X , X)

pi Sazonov - pi pi pi pi. pi pi pi R.A Minlos.

6.2.5.

1. = (1, ..., n) Rn |ni=1

aii| 1 a =

(a1, ..., an) Rn mi=1

ia2i 1 pi m n i > 0 i = 0

i = m+ 1, ..., n.

2. = (1, ..., m) Rm |mi=1

ii| 1 = (1, ..., m)

mi=1

2i mi=1

2i 1 ( > 0) .

pi.

1. m < n. k 6= 0 k > m. a Rn ai = 0 i 6= k ak = 2|k|

mi=1

ia2i = 0 1

|ni=1

aii| = |akk| = 2 > 1 - pi.

2. , mi=1

2i >1 i =

i

|| mi=1

2i =

mi=1

ii =|| > 1

= 1

103

mi=1

2i 1 Rm mi=1

2i

pi Cauchy-Schwartz |mi=1

ii| 1.

6.2.6. , pi (Rn,Bn) pi Gauss . (0, 1) :

({x Rn : (x) }) 11

Rn

(1 (t))d(t)

pi. pi pi 1 0pi

Rn

(1 (x))d(x) A

(1 (x))d(x) pi A = {x : 1 (x) 1 }.pi

Rn

(1 (x))d(x) (1 )(A).

Rn

(1 (x))d(x) = 1Rn(x)d(x)

= 1Rn

(

Rnei(x,y)d(y))d(x)

= 1Rn

(

Rnei(x,y)d(x))d(y)

= 1Rn(y)d(y)

=

Rn

(1 (y))d(y)

pi A = {x Rn : (x) } pi .

6.2.7. pi pi pipi pi pi

Rn

(1 (y))d(y) pi

Rn

(1 (x))d(x).

6.2.8. (Minlos) pi Rn pi pi > 0 pi

|1 (x)| + (Bx, x) x Rn (1)

pi B : Rn 7 Rn , .pi A : Rn 7 Rn , {x Rn : Ax =

104

0} {x Rn : Bx = 0}, - 1, ..., m e1, ..., em. r > 0 :

(Rn \ E) 3(+ rmk=1

(Byk, yk))

pi yk = 1k ek (k = 1, ...,m), E = {x Rn : (Ax, x) r} E = {y Rn :

|(y, x)| 1 x E} -pi .pi. r = 1.pi {e1, ..., em} em+1, ..., en {e1, ..., en} pi Rn. Aek = 0 pi Bek = 0 k =m+ 1, ..., n. pi pi {e1, ..., en} :

(Ax, x) =

mj=1

j(x, ej)2 =

mj=1

2j (x, yj)2

(y, x) =

ni=1

(y, ei)(x, ei)

pi pi E E E = {y Rn : |ni=1

(y, ei)(x, ei)|

1 x Rn mi=1

j(x, ej)2 1}. pi 6.2.5. (1)

pi y Rn pi E (y, ei) = 0 i = m + 1, ..., n. pi ei =

iyi, i = 1, ...,m pi-

E = {y Rn : |mi=1

i(y, yi)(x, yi)| 1 x Rn mi=1

2i (x, yi)2 1

ni=m+1

(y, ei)2 = 0}

pi 6.2.5. (2) pi

E = {y Rn :mi=1

(y, yi)2 1

ni=m+1

(y, ei)2 = 0} (1)

k N Ak : Rn 7 Rn

Akx =

mi=1

(x, yi)yi + k2

ni=m+1

(x, ei)ei

(Aky, y) =

mi=1

(y, yi)2 + k2

ni=m+1

(y, ei)2 (2)

105

pi (1),(2) pi

E =k=1

{y Rn : (Aky, y) 1} (3)

k pi Gauss 0 - Ak k(x) = exp{ 12 (Akx, x)}, x Rn.pi x Rn x =

ni=1

(x, ei)ei

(Bx, x) (Bx, x) =n

i,j=1

(x, ei)(x, ej)(Bei, ej)

Rn

(Bx, x)dk(x) =

ni,j=1

(Bei, ej)

Rn

(x, ei)(x, ej)dk(x)

=

ni,j=1

(Bei, ej)(Akei, ej)

pi (2)

(Akei, ej) = 0 i 6= j (Akei, ei) ={

(ei, yi)2 , i m

k2 , i > m

pipi Bei = 0 i = m+ 1, ..., n Rn

(Bx, x)dk(x) =

mi=1

(Bei, ei)(ei, yi)2

ei =iyi i = 1, ...,m

Rn(Bx, x)dk(x) =

mi=1

(Byi, yi) (4)

pi k pi

({x : k(x) < e 12 ) = ({x : (Akx, x) > 1}) pi 6.2.6. ( = e

12 )

({x : (Akx, x) > 1}) 1

Rn

(1 (t))dk(t)

pi pi pi

1Rn

[+ (Bt, t)]dk(t) pi = e12 pi 1 3

(4)

({x Rn : (Akx, x) > 1}) 3(+mi=1

(Byi, yi)) (5)

106

pi (3),(5)

(Rn \ E) = (k=1

{x Rn : (Akx, x) > 1})

= limk

({x Rn : (Akx, x) > 1)

3(+mi=1

(Byi, yi))

6.3 To jerhmaMinlos kai to jerhma Sazonov

pi pi - pi , pi .

6.3.1. Hilbert H S ,, pi H. S pi pi S = u S1 u pi S1 : H 7 H pi , pi u : H 7 H (, ) pi.pi. pi pi pipi pi , A : H 7 H pi S = A A (pi S). pi (pi) S pi

nn }}}}}}}

v// H

w

OO

S1 = v v pi S1 : H 7 H , pi. ([15] .224). pi w : H 7 H pi S = w S1 w. w = u .

6.3.2. pi ... X pi X = (X, )

. R R(X , X).

107

R = w S w S S(H) w : H 7 X , pi pi ( ) C (X,X ) pi ( ) pi pi pi norm Hilbert H.

pi. pi R(X , X) pi Hilbert H , S1 S(H) , ( ( ) )v : H 7 X R = v S1 v. v pi (H,H ) (X,X ) H pi B = {h H : h 1} H (H,H )-pi v(B) (X,X )-pi. pi v(B) ,pi ( ) pi C (X,X )-(Robertson Robertson . 71). pi pi v : H 7X ( )C (X,X ) . 6.3.1. S1 S1 = u S u pi S S(H) u : H 7 H , pi. pi w = v u w , pi R = w S w.

6.3.3. X pi ... pi X = (X, ). :

1. pi C(X,X ) pi .. : X 7 C pi Sazonov S(X , X). pi pi C (X,X )- X.

2. X : X 7 C X (0) = 1 pi pi Sazonov S(X , X). pi C (X,X )- X pi = X . .

pi.

1. > 0. pi X piS(X

, X) pi pi V = {x X :< Rx, x >< 1} R R(X , X) :

|1 (x)| < 6

x V

pi |1 (x)| 2 x X |1 (x)| 2 < Rx, x > 1

|1 (x)| < 6

+ 2 < Rx, x > x X (1)

6.3.2. R R(X , X) pi -pi R = uSu pi S S(H), H Hilbert u : H 7X , pi pi ( ) C (X,X ).

M = {u(x) : x r}

108

pipi M X C (X,X )-pi. n N {x1, ..., xn} X f : X 7 Rn f(x) = (x1(x), ..., x

n(x)) a = f u b = a

S

A = aa B = bb

A,B : Rn 7 Rn , pi{y Rn : Ay = 0} {y Rn : By = 0}pi 1, ..., m - e1, ..., em A yi = 1i ei, i = 1, ...,m {ay1, ..., aym} H (pi )

(ayi, ayj) = (aayi, yj) = (Ayi, yj) =1

(Aei, ej) = (ei, ej) i = 1, ...,m.

pi ( 44 pi)

E f(M) (2)pi E = {y Rn : (Ay, y) 1r2 } E pi . vf (Rn,Bn)

vf (B) = (f1(B)) , B Bn

y f = y1x1 + ...+ ynxn y Rn pi

(y f) = vf (y) , y Rn

pi (1)

|1 vf (y)| 6

+ 2 < Ry f, y f > , y Rn

pi ( 45)

< Ry f, y f >=< By, y > y Rn (3)

|1 vf (y)| 6

+ 2 < By, y > , y Rn

pi pi pipi Minlos pi vf A,B pi

vf (Rn \ E) 3( 6

+2

r2

mk=1

(Byk, yk)) (4)

mk=1

(Byk, yk) =mk=1

(aSayk, yk) =mk=1

(S(ayk), ayk) trS pi pipi {ayk, k = 1, ...m} H . pi pi (4)

vf (Rn \ E) 2

+6

r2trS

109

pi r > 0 pi r2 > 12 trS pi (2) ,

vf (Rn \ f(M)) < n N f = (x1, ..., xn), xi X .pi 41 pi Prohorov 4.4.11. K = M pi C (X,X )-pi = X C(X,R). pi pi ... (X,C (X,X )).

2. pi (X , S(X , X)) pi - X : X 7 C , , : pipi pi X . pi 5.2.4. ( = X ) pipi X .. = X (pi) pi Sazonov. pi (1).

6.3.4. 4.4.1. C (X,X )- -.

44. pi (2) pi.

Updeixh: Ap to Lmma 6.2.5. kai pwc akribc gia to Lmma Minlos E = {x Rn :

mi=1

(x, yi)2 r2 kai

ni=m+1

(x, ei)2 = 0} 'Etsi tuqn x E grfetai x =

mi=1

(x, ei)ei. Qrhsimopointac tic Aei = iei kai A = aaqoume diadoqik: x =

mi=1

12i

(x, aa(ei))aa(ei) =mi=1

12i

(a(x), a(ei))aa(ei) = a(h) pou h =mi=1

12i

(a(x), a(ei))a(ei)

kai afo ei =iyi to h =

m1

1i

(a(x), a(yi))a(yi). 'Omwc ta a(yi), i = 1, ...,m enai

orjokanonik kai ra h2 =m1

1i

2(a(x), a(yi))2 =

m1

12i

(x, aa(yi))2 =m1

12i

(x,A(yi))2 =

m1

12i

(x, iyi)2 =

m1

(x, yi)2 r2 afo x E. 'Wste x = a(h) me h2 r2. 'Omwc

a = f u.

45. pi (3) pi. pi a = f u aSa = B pi B.

Updeixh: Gia y = (y1, ..., yn) Rn kai f = (x1, ..., xn) enai < Ry f, y f >=i,j

yiyj < Rxi, xj >. Exllou < By, y >=

i,j

yiyj < Bvi, vj > pou {vi} h sunjhcbsh tou Rn.Ja dexoume tra ti u(xi) = a(vi) gia i = 1, ..., n. Prgmati hsqsh aut isoduname me (x, u(xi)) = (x, (f u)(vi))x H < u(x), xi >=

110

((f u)(x), vi)x H xi(u(x)) = xi(u(x))x H pou alhjeei. Sunepc -qoume < Rxi, x

j >=< uS(u

(xi)), xj >=< uSa

(vi), xj >=< uSa(vi), pij f >=

pij f(u Sa(vi)) = pij(aSa(vi)) = pij(Bvi) =< Bvi, vj > pou pii : Rn 7 R hsunjhc i-probol.

(2) pipi ; pi . pipi pi X Hilbert . .

pi X Hilbert. pipi R(X , X) pipi , pi R : X 7 X pi pipi S(X). pi Sazonov pipi Hilbert X pi S(X) pi pipi pi pi x 7 (Rx, x), R S(X). Sazonov pi pipi pi pi.

6.3.5. pi Hilbert X pipi pi

X

x2d(x) < +. S : X 7 X pi pi

r(x, y) =

X

(u, x)(u, y)d(u)

pi (Sx, y) = r(x, y) x, y X. 6.3.6. pi Hilbert X pipi

X

x2d(x) < +. S , pi.

pi. pi pi pi ( 6.1.3.). {ei, i I} X. pi pi-pi pi ein , n N

n

(Sein , ein) =n

X

((u, en)2d(u)

X

u2d(u) < pi {i : (Sei, ei) > 1n}, n N pipi (Sei, ei) > 0 pi i I.pipi

iI

(Sei, ei) =i

(u, ei)

2d(u) = u2d(u) < . pi

S pi.( [16] . 161 )

6.3.7. (Sazonov ) X Hilbert. X : H 7 C

111

pi (X,BX) X X (0) = 1 piSazonov S(X).

pi. pi pi. . .. (x) =

X

ei(x,t)d(t), x X x = 0 pi S(X). > 0 pi K = K X (K) > 1 4 . (X,B(X)) (B) =

(BK)(K) . (K) = 1 pi

X

u2d(u) < +.

|1 (x)| |(K)K

ei(x,t)d(t)|+Kc|1 ei(x,t)|d(t)

pi (L) = (K)(L) L K |1 ei(x,t)| 2 t X pi

|1 (x)| |(K) (K)K

ei(x,t)d(t)|+ 2(Kc)= (K)|1 (x)|+ 2(Kc)

(Kc) < 4 , (K) 1

|1 (x)| < |1 (x)|+ 2x X (1)

pi |1 eiy| < 2|y|, y R pi x X

|1 (x)| X

2|(x, t)|d(t)

4(x, t)2d(t) (2)

S pipi S(X) (Sx, x) =

X

(x, t)2d(t) pi pi

(1),(2) pi

|1 (x)| < 4(Sx, x) + 2x X

pi pi Sazonov V = {x X : (Sx, x) 0 pi A A (x)(x(A)) > 1 x X .

A (X,X )- pi X pi pi pi:

. A A > 0 A A

113

. A,B A pi E A A B E. A A pi pi B A A B. A A A(x) = sup{|x(x)| : x A}, x X. A seminorm X seminorm (A, A A ) pi pi X . pi pi- A -pi A (X , X). A (X , X) C (X , X).

6.3.11. X pi ... X . pi (X,C(X,X )) A (X,X )- pi X pi pi (),(),() pi pipi. pi pi :

1. pi A

2. A (X , X).

pi. [15] . 193 [16] . 412

6.4 Mtra pijanthtac kai q.. se dukoc

qrouc

X pi ... X . X pi

C(X , X) =iIA i

pi I = {(x1, ..., xn) : n N, xi X} i = (x1, ..., xn) I - A i = {g1(B) : B Bn} g(x) = (< x1, x >, ..., < xn, x >), x X . - C(X , X) = (

iIA i ).

pi C(X , X) - X

(x) =

Xeid(x) , x X

(X,A x, ) x X. pi .. pipi pi - pi C(X , X). R(X,X ) R : X 7 X pi pi Hilbert H v : X 7 H R = vSv pi S S(H).

114

pi Sazonov S(X,X ) X pi pipi pipi pi seminorm {R, R R(X,X )} pi R(x) =< x,Rx >, x X. 6.3.3. :

6.4.1. X pi .. X .

1. (X , C(X , X)) pi .. (x)x X pi Sazonov S(X,X ). pi pi C (X , X)- X .

2. X : X 7 C X (0) = 1 pi Sazonov S(X,X ). pi C (X , X)- X pi = X . .

pi.

1. Y = (X , (X , X)) pi (X , X) pi Mackey X . Mackey-Arens Y Y X pi .. Y . pi pi SazonovS(Y

, Y ) pi 6.3.3. (Y, Y ). X X pi Mackey (X,X ) pi Y Y X pi R(Y , Y ) = R(X , X ) pi S(Y , Y ) = S(X , X ). pi S(X , X ). R(X,X ) R(X , X ) R R(X,X ) R = vSv piv : X 7 H pi X ( ) Hilbert H pi (X,X ) ( ) pi X Mackey. pi pipi - pi S(X,X ) S(X , X ) pi pi S(X,X ) S(X , X ),pi pi pi .

2. pi (2) 6.3.3. pi pi- 5.2.4. (pi ).

Minlos ( X ) pi X pi pi (nuclear).

6.4.2. E,F Banach u : E 7 F , . u pi pi {xk, k N} E {yk, k N} F

k

xkyk yk , x E.

115

pi pipi pi E = F = H pi H Hilbert pi pipi 6.2.1.

6.4.3. pi .. X pi T pi X Banach E pi Banach F, , u1 : X 7 F pi u2 : F 7 E pi T = u2 u1.

pi ( Schaeffer, H. Topological Vector Spaces ) pi pi pipi pi Sazonov S(X,X ) pi pi pi pipi pi:

6.4.4. (Minlos ) pi X X . :

1. pi (X , C(X , X)) pi .. pi pi C (X , X)- X .

2. X : X 7 C , X (0) = 1 piC (X , X)- pi pi = X . .

6.4.5. X pi X . X : X 7 C .. C (X , X)- pi- X X , X (0) = 1.pi. pi X (2) pi pi pi X pipi pi- Sazonov . pipi pi xn x < xn, x

>< x, x > x X . pipi |ei| = 1 (xn) (x). pi X pi X = .

6.4.6.

1. pi.. D(V ),D (V ),E (V ),E (V ),(V ),(V ), V Rn . Hilbert H pi S(H) pi.( [16] . 411)

2. pipi pipi pi X barelled (tonelle). [18].

116

Keflaio 7

Mtra Pijanthtac Gauss

7.1 Mtra Pijanthtac Gauss ston Rn

(Rn,Bn) pi

(B) =

B

(2pi)n2 e

12 |x|2dx ,B Bn

pi ..

(t) = e12 |t|2 t Rn

7.1.1. a Rn - nn-pi . Gauss Rn pi a, (Rn,Bn) pi

(B) = (T1(B)) , B Bn

pi T (x) = a + 12x , x Rn. Gauss

pi a = 0 = I.

pi pi 1 1 pi T1(B) =

12 (B a).

pipi x = 12 (y

a)

(B) =

T1(B)

(2pi)n2 e

12 |x|2dx

=

B

(2pi)n2 (det )

12 e

12 (1(ya),ya)dy

Gauss pi a, pi

d(y) =1

(2pi)n2 (det )

12

e12 (1(ya),ya) , y Rn

117

( (x, x) = 0 pi x 6= 0) Gauss pi pi Lebesgue (- pi A T (Rn) = Lebesgue pi (A) = (T1(A)) = () = 0). pi pi

12 -

Rneit((z)+a)d(z) =

eityd(y) = (t) , t Rn

pi (5) pi 5.2.3. t(z) = zT (t) pi pi

eitaRneit(z)d(z) = eita

Rneiz

T (t)d(z)

pi (t) = eita(T (t)) , t Rn. (T (t)) = e

12 |T (t)| = e

12 (t,t)

(t) = eita12 (t,t) , t Rn (1)

pi .. Gauss pi, . n = 1 = () 0 a R Gauss pi a,

(B) =

{a(B) = 0B

12pi

e12 (xa)2dx > 0

pipi (n = 1) pi Ryd(y) = a ,

Ry2d(y) a2 = .

7.1.2. pi - pi .. pi pi- Gauss pi a, pi pi .. pi (1) pipi.

7.2 Mtra pijanthtac Gauss se dianusmatikocqrouc aperwn diastsewn

X pi .. X . pi - pi pi C(X,X ) pi C(X,X ) X pi (R,B1) pi

(A) = (1(A)) , A B1 (1)

118

7.2.1. pi (. pi) Gauss (. Gauss) pi ... X - A C(X,X ) (. - C(X,X )) pi pi pi: X pi (1) Gauss R.

7.2.2.

1. Gauss y2dx(y) < +

x X X

| < x, x > |2d(x) < + x X . pi .

2. Gauss pi C(X,X ) Gauss . pi pi pi. pi - Gauss .. pi pi:

(x) =X

eid(x) , x X (2)

pi pipi - (2) (X,(x), ).

7.2.3. pi .. X X . pi - A C(X,X ). : Gauss X pi a(x), x X Q(x), x X pi

(x) = eia(x) 12Q(x) , x X (3)

pipi a(x) =X

< x, x > d(x)

Q(x) =X

| < x, x > |2d(x) a2(x) x X .

pi. pi

(yx) = x(y) y R, x X (4) Gauss x Gauss R1 pim =

Rydx(y) =

Ry2dx(y)m2 ..

x(y) = eimy 12y2 , y R

pi (pi ) m =X

xd, =X

(x)2dm2 pi (4) y = 1 pi (3)

119

a(x) =X

xd Q(x) =X

(x)2d a2(x), x X . pi (3) x (pi (4) )

x(y) = (yx) = eia(x

)y 12Q(x)y2 , y R pi x Gauss R1 pi a(x) Q(x). a(x) =

Rydx(y) Q(x) =

Ry2dx(y) a2(x) pi (

) a(x) =X

xd Q(x) =X

(x)2d a2(x).

7.2.4. pi pi Gauss - pi pi .. pi (3). pi pi a,Q. - Gauss pipi pi ..

X Hilbert X = X :

7.2.5. X Hilbert pi - A C(X,X ). Gauss (x) = eia(x)

12Q(x), pi a Q X.

pipi a(x) =X

(y, x)d(y) Q(x) =X

(y, x)2d(y) a2(x), x X.

pi pi pipi pi pi:

7.2.6. pipi pi pi , - A pi C(X,X ) Gauss pi - Gauss .

Gauss pi pi pi:

7.2.7. pi ... X X . a : X 7 R Q : X 7 R. pi Gauss pi a,Q.

pi.

X (x) = eia(x) 12Q(x) , x X

X (0) = 1. X (x), x X . pi ( [16] . 187)

120

eia(x), e

12Q(x

), x X . {c1, ..., cn} C {x1, ..., xn} X

nk,`=1

ck cèia(xkx`) =

nk,`=1

ckeia(xk) cèia(x`) =

nk=1

ckeia(xk)

2

0

b , Q(x) = b(x, x), x X . ( b(x, y) = 14 [Q(x

+ y)Q(x y)]). b (b(x, x) 0 x X ) {1, ..., n} R {x1, ..., xn} X

ni,j=1

ijb(xi, xj) = b(

ni=1

ixi,

ni=1

ixi) 0 (

i R). pi pi Q(x) = b(x, x) Q(xk x`) = Q(xk) + Q(x`) 2b(xk, x`) pi pi A =n

k,`=1

kè 12Q(xkx`) =

nk,`=1

ke 12Q(xk)è 12Q(x`)eb(xk,x`) me 12Q(xm) =

m A =n

k,`=1

kèb(xk,x

`). -

e ([16] . 187) pi A 0. X ( pipi pi X ). pipi piE X {e1, ..., ek} . pi pi E pi norm |x| = (y21 + ... + y2k)

12 x = y1e1 + ... + ykek pi

xn = yn1 e1 + ... + y

nk ek , n N x = y1e1 + ... + ykek

xn x E ynm ym m = 1, ..., k.pi a(xn) =

km=1

ynma(em) k

m=1yma(em) = a(x

) xn x E. b X pi

Q(x) = b(x, x) Q(xn) = Q(k

m=1ynmem) =

ki,j=1

yni ynj b(ei, ej) pi

Q(xn) k

i,j=1

yiyjb(ei, ej) = Q(k

m=1ymem) Q(xn) Q(x).

X (xn) X (x) xn x E. X pi pipi 5.2.4. pi X pi = X . pi 7.2.6. pi Gauss pi a,Q.

7.2.8. X Hilbert pi

121

X (x) = e 12x2 , x X. pipi pi Gauss X pi = X . pi pi pi - A C(X,X ). pi 5.1.5. .. = X pi X X (xn) X (x) xn, x X (xn, y) (x, y) y X pi xn x (xn, y) (x, y) y X. xn X pi x X xn x.

7.3 To q. . mtrwn Gauss se qrouc Hilbert.Jerhma Mourier

pi pipi pi Gauss pi , :

X

| < x, x > |2d(x) < + x X

pipi pi X Hilbert -

X

(u, y)2d(u) < + y X

pi pi X

|(u, y)|d(u) < +

X

|(u, x)(u, y)|d(u) < + x, y X pi

r(x, y) =

X

(u, x)(u, y)d(u)X

(u, x)d(u) X

(u, y)d(u)

r , ( pi Riesz ) R : X 7 X pi

(Rx, y) = r(x, y) x, y X (1)

R , ((Rx, x) 0 x X) X. pi (1)

(Rx, x) =

X

(u, x)2d(u)(

X

(u, x)d(u)

)2, x X

pi .. Gauss ( 7.2.5.)

(x) = eia(x)12 (Rx,x) , x X (2)

122

pi a(x) =X

(u, x)d(u), x X.pi ( Hilbert) pi m X pi

a(x) = (m,x) x X(pi Pettis m =

X

xd(x))

.. Gauss

(x) = expi(m,x)12 (Rx,x) , x X

m X .

pi pi pi pi pi .. Gauss Hilbert.

7.3.1. (E. Mourier ) X Hilbert pi (X,B(X)). Gauss ..

X (x) = ei(m,x) 12 (Rx,x) , x X (3)pi m X R , pi (nuclear ) X.pi. Gauss. pi- .. (3) pi m X R . R pi. Sazonov .. x = 0 pi Sazonov S(X). pi > 0 pi S S(X) :|1 (x)| < 1 e 12 x X (Sx, x) < 1. 1 Re(x) |1 (x)| Re(x) e 12 (Rx,x) pi 1 e 12 (Rx,x)

(Sx0, x0) (S

x0d ,

x0d ) =

1d2 (Sx0, x0) < 1 pi (4) pi (R

x0d ,

x0d ) < pi

(Rx0, x0) < d2 d >

(Sx0, x0)

x0 X(Rx0, x0) (Sx0, x0) (5)

pipi {ej , j I} pipi pi- {ejn , n N}

n

(Rejn , ejn) n

(Sejn , ejn) < +

123

S pi.pi 6.3.6. pi (Rej , ej) > 0 pi j I. pi pi pi (5)

j

(Rej , ej) 0 pi V = {x : (Rx, x) < 2 ln(1 )} pi |1 h(x)| < x V . h pi pi Sazonov pi v pi v = h. (X,B(X)) (B) = v(1(B)) pi(x) = x+m (B) = v(B m). :

(x) =

X

ei(x,y)d(y)

=

X

ei(x,y+m)dv(y)

= ei(x,m)v(x)

= X (x) x X

124

Parrthma A

Apdeixh tou Jewrmatoc

2.4.2.

.0.2. pi Hausdorff (X, T ) U1, U2 T . piK K K U1U2. pi piK1,K2 K K = K1 K2 K1 U1 , K2 U2.

pi. K U c1 , K U c2 pi (X Haus-dorff) pi V1, V2 V1 K U c1 ,V2 K U c2 V1 V2 = . K1 = K V c1 K2 = K V c2 . K1,K2 pi .

.0.3. H pi S H. : H 7 [0,] () = 0 pi pi (H, ). M = {X S : (A X) + (A Xc) = (A) A S} - pi S. X M (A) (A X) + (A XC) A H (A) 0 pi pi {An, n N} H

nAn T

n=1

(An) < (T ) + (1)

pi T X n=1

AnX T Xc

nAn Xc : (T X)

125

n=1

(An X) (T Xc) n=1

(An Xc)

(T X) + (T Xc) n=1

[(An X) + (An Xc)] (2)

pi (1) (An) 0

(T X) + (T Xc) n=1

(An) < (T ) +

(T ) (T X) + (T Xc) T S.

Apdeixh tou Jewrmatoc 2.4.2: : T 7 [0,]

(U) = sup{(K) : K K K U} , U T pi pi (T , ).

(U) (U) U T1. M T

0.3. (A) (AB)+(ABc) A,B T (A) < +. : D K D A B pi: E K E A DC

D E K , D E A D E = (A) (D E) = (D) + (E)( pi pi K).pi pi

sup{(E) : E K E A Dc} (A) (D) pi A DC T pi

(A) (A Dc) + (D) D K D A B (?) D K D A B : A Dc A Bc A Dc T pi (A Bc) (A Dc) pi (?)pi

(A) (A Bc) + (D) D K : D A Bpi sup{(D) : D K D A B} = (A B) (A) (A Bc) + (A B). (A B) (A B) pi .

126

2. -pipi T . pipi . pi pi-pi -pipi T . : U1, U2 T . K U1 U2 0.2.pi pi K1,K2 K1 U1,K2 U2 K = K1 K2. :

(K) = (K1 K2) (K1) + (K2) (U1) + (U2)

pi sup{(K) : K K K U1U2} (U1)+(u2). pi (U1 U2). {Un, n N} T . K K K

n=1

Un

pi pipi pi, K mk=1

Unk pi (K)

(mk=1

Unk). (pi) pipi (mk=1

Unk) mk=1

(Unk) pi (K) n=1

(Un) pi K n=1

Un. .

3. (U) = (U) U T . (U) (U), U T . pipi (U) < +. :

(U) = inf

{ n=1

(Vn) : Vn T n=1

Vn U}.

-pipi , pipi

{Vn, n N} T n=1

Vn U

n=1

(Vn) (n=1

Vn) (U).

(U) (U) U T .

4. (A) = inf{(U) : U T U A} A X.

(A) = + .

(A) < + > 0 pi {Un, n N} T n=1

Un A

(A) n=1

(Un) < (A) +

127

V =n=1

Un V A, V T

(A) (V ) n=1

(Un) n=1

(Un)

pi (A) (V ) < (A) + 5. (K) = (K) K K

K K. 3,4

(K) = inf{(U) : U T U K}

pi (U) (K) U T ,K K U K. (K) (K). > 0 pi pi pi U T U K (C) < (K) + C K K C U. pi :

(U) = sup{(C) : C K K C U} (K) +

pi 3 (U) (K) + . U K (K) (K) + > 0

128

Parrthma B


2.4.4.

pi pi pi- . pi {A, } :

pipi Aa, Ab pi pi A pi AaAb A .

pi pi pi pipipi i, ii, iv.

.0.4. {U} A

U = X.

sup(U) = 1.

pi. > 0 pi K X pi pi iv. pi pipi I K

I

U A pi (pi pipi pi ii)

1 (I

U) > 1

pi I

U U A pi

1 (U) > 1 .

U TX (U) = sup{(B) : B U,B A }

pi pi (, TX).

129

1. {U} A U =

U.

(U) = sup(U)

pi (U) (U) sup(U) < (U) pi.

> 0 pi

(U) sup(U) > 2 (1)

pi pi B A B U

(B) > (U) 2

pi pi iii pipi F A F B

(F) > (B) 2

pi(F) > (U) (2)

pi {U F c } A

(U F c ) = X (pi )

sup(U F c ) = 1 (3)

(2), (1)

sup(U F c ) sup

[(U) + 1 (F)]

sup(U) + 1 (U) +

< 2+ 1 + = 1 < 1 pi

2. {U} TX

U = U

(U) = sup(U)

H = {A X : A A , A U pi }. H

AH

A = U .

x U pi x U pi Api U =

iI

Ai Ai A , pi x Ai

130

A U. A H . pi. 1

(U) = sup{(A) : A H } (4)

pi A H A U pi pi(A) (U) pi , (4) sup

(U).

(U) sup(U)

pipi pi (U) (U) .

3. (pi) pi, pipi -pipi. U1, U2 TX U1 U2 = . A pi pi

U1 =aI

V 1a V1a A ,

pi

U1 =

(ai

V 1a : i pipi I) =i

U1i

{U1i } A T pi . U2 =

j

U2j {U2j } A T pi

U1i U2j = i, j

{(i,j) = U1i U2j } A T pi(i,j)

(i,j) = U1 U2 pi 1

(U1 U2) = sup(i,j)

(U1i U2j )

= sup(i,j)

[(U1i ) + (U2j )]

= supi(U1i ) + sup

j(U2j )

= (U1) + (U2)

-pipi, Un TX , n N.

(n=1

Un) = ({ni

Un : i pipi. N})

pi 2 pipi (pi

131

pi pi pi)

(n=1

Un) = sup{(ni

Un) : i pipi. N}

supi

ni

(Un)

n=1

(Un)

4.(U) = (U) U TX

(U) (U) U X, pi. {Vn, n N} TX

nVn U TX pi 3

n

(Vn) (n

Vn) (U)

pi pi (U) (U).

5. > 0 pi pi K : (K) > 1 .pi pi iv pi pi K X

(B) > 1 B A B K (5)

Kc A pi pi Kc =

i

Ui {Ui} A T . pi 1,4

(Kc ) = (Kc )

= supi(Ui)

= supi

(1 (U ci ))= 1 inf

i(U ci )

pi pipi

(Kc ) 1 (K)

pi(K) inf

i(U ci )

U ci A U ci K. pi (5).

132

6.M B(X)

U T (A) (A U) + (A U c) A T

pi pipi pi U T pipi U c = L. pi x U pi pi Nx x NL L Nx NL = . pi pi Nx U Nx A A pi pi. Nx N cL = N cL U piU =

xU

Nx Nx A Nx U . U =

V V A V U pi

U =

(i

V : i pipi. ) =i

Ui

pi {Ui} A Ui = Fi U . A = (A Ui) (A U ci ) (A Ui) (A F ci ). pi 3 4

(A) [(A Ui) (A F ci )]= (A Ui) + (A F ci ) (A Ui) + (A F ci ) i

F ci U c

(A) (A Ui) + (A U c) i pi 2 4

supi(A Ui) = (A U) = (A U)

pi

(A) (A U) + (A U c) A TX U TX U c pi M pi M KX . pi 5 pi pi Kn X,n N (Kn) > 1 1n pi Y =

n=1

Kn Y M (Y ) = 1. U T F = U c F =(F Y ) (F Y c). pi (F Y c) (Y c) = 0 F Y c M F Y =

n=1

(F Kn) F Kn KX .

133

7.(A) = inf{(U) : U T U A} , A X

> 0 pi {Un, n N} TX nUn A pi

(A) n

(Un) < (A) +

U =n=1

Un U TX , U A

(A) (U) < (A) +

8. = A B A . U TX U B

(U) = (U) = sup{() : U, A } (B)

piinf{(U) : U TX U B} (B)

pi 7 pi (B) (B). pi ii, iii

(B) = inf{(U) : U A , U U B}

> 0 pi U A TX

U B (U) < (B) +

pi A

() < (B) + A Upi , 4 U B pi > 0

(B) (U) = (U) (B) +

(B) (B) (X,M) pipi pi 5,6,7 pi 2.4.9. pi v. pi 2.2.5. ( 14).

134

Parrthma G


Prohorov

pi :

.0.5. KX pi pi. X :1. K =

iI

P1i (Pi(K)) K KX

2. K,L KX K L = pi i I Pj(K)Pj(L) = j I i j.

3. Pj(K) P1ij (Pi(K)) i j I K KX .pi.

1. K P1i (Pi(K)) K iI

P1i (Pi(K)). pi

K 6= ( pi) x iI

P1i (Pi(K)).

Pi(x) Pi(K) i I pi Kxi = {y K :Pi(y) = Pi(x)} , i I - K Kxi ( Pi ) pi pi. pi i j Pj(y) = Pj(x) (Pij Pj)(y) = (Pij Pj)(x) Pi(y) = Pi(x) Kxj Kxi i j. pi , pi Kxi , i I Kxi 6= pi

iI

Kxi 6= pipi y X y Kxi i I pi Pi(y) = Pi(x) i I. {Pi, i I} x = ypi x Kxi K.

2. Mi = P1i (Pi(K))L, i I. pi K,L

Pi pi Mi, i I pi. pipi i j :(Pij Pj)(K) = Pi(K) pi Pj(K) P1ij (Pi(K)) pi

135

P1j (Pj(K)) P1j (P1ij (Pi(K))) = (Pij PJ)1(Pi(K)) = P1i (Pi(K)) Mj Mi i j.pi pi (1)

iI

Mi = KL = mk=1

Mik = {i1, ..., im} I. pi I pi pi i I i1, ..., im i Mi = . j I i j Mj = . P1j (Pj(K)) L = Pj(K) Pj(L) =

3. i j I (Pij PJ)(K) = Pi(K) P1ij (Pij(Pi(K))) =P1ij (Pi(K))

Pi(K) P1ij (Pi(K))

.0.6. I pi pi ai 0 bi 0 :i j aj ai bj bi.

infiI

(ai + bi) = infiI

ai + infiI

bi

pi. infiai = a, inf

ibi = b > 0 pi i, j I

a ai < a+ 2 b bj < b+ 2 . pi I k I i, j k pi ak ai bk bj pi

a+ b ak + bk a+ b+ infi{ai + bi} = a+ b


1. Pi(K) pi Xi K KX i I. : KX 7 [0,)

(K) = inf{i(Pi(K)) : i I}

pi 0.5. pi

j(Pj(K)) j(P1ij (Pi(K))) = i(Pi(K))

i j I K KX pi

(K) = inf{j(Pj(K)) : i j} (1)

pi pi 2.4.2.:

K L KX (K) (L). .

136

K,L KX K L = 0.4. pi i I :Pj(K)Pj(L) = i j pi (1) :

(K L) = inf{j(Pj(K L)) : i j}= inf{j(Pj(K)) + j(Pj(L)) : i j}= inf{j(Pj(K)) : i j}+ inf{j(Pj(L)) : i j}.

(K L) = (K) + (L). pi K,L KX

(K L) (K) + (L).

> 0 K KX . pi i I pi

(K) i(Pi(K)) < (K) + 2

(2)

i pi pi U Xi U Pi(K)

i(U) < i(Pi(K)) +

2(3)

W = P1i (U) W TX ,W K C KX K C W Pi(C) Pi(W ) = Pi(P1i (U)) U i(Pi(C)) i(U) pi

(C) i(U) (4)

pi (2),(3),(4) C KX K C W

(C) < (K) +

pi W X W K. pi 2.4.2. pi (X,B) (K) = (K) K K. pipi

(P1i (B)) = sup{(K) : K pi P1i (B)} (5)

i I B Bi. K P1i (B) Pi(K) Pi(P

1i (B)) B pi

i(Pi(K)) i(B) pi (K) = (K) i(B).pi (5) pipi

(P1i (B)) i(B) i I, B Bi

137

2. pi i, j I k I i, j k k(P

1ik (Xi)) = i(Xi) i(Xi) = k(Xk). j(Xj) =

k(Xk) pi pi = i(Xi), i I. K KX

(K) = infiI

i(Pi(K))

= infiI

i [Xi \ (Xi \ Pi(K))]= infiI

( i(Xi \ Pi(K)))= sup

iIi(Xi \ Pi(K)).

pi

(K) + supiI

i(Xi \ Pi(K)) = ,K KX (6)

pi Prohorov. > 0pi pi K KX sup

iIi(Xi \ Pi(K))

pi (6) pipi (K). pi pi (1) (K) (X) = (P1i (Xi)) i(Xi) = pi

(K) = sup{(K) : Kpi X} (X) = . i I,B Bi

(P1i (B)) = i(B)

vi(B) = (P1i (B)), B Bi. vi (Xi,Bi)

pi vi(Xi) = i(Xi). pipi pi (1) vi i. 2.2.6. vi = i. pi . pi (P1i (B)) =i(B) i I,B Bi. K KX 0.5. K =

iI

Fi Fi = P1i (Pi(K)), i I. Fi

i j Fi Fj ( pi ). pi(K) = inf

iI(P1i (Pi(K))) = inf

iIi(Pi(K)) = (K) K KX

, = . . pi (P1i (B)) = i(B) i I,B Bi Prohorov. = (X) = i(Xi), i I. (6) pi (K) + sup

iIi(Xi \ Pi(K)) = K KX pi

K KX(K) + sup

iIi(Xi \ Pi(K)) = (X) (7)

138

pi > 0 K KX (X) < (K) + pi pi (7)

supiI

i(Xi \ Pi(K)) < .

139

Bibliografa

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[12] , . - pi, . : ,

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[15] Schwartz , L. : Radon Measures , Oxford.

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141

[19] Neveu , J. : Calcul de Probabilites , Paris 1970

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142

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