08_Abstract Construction of Probability Measures-NEW (1)

69

8.

, , . ( ). , -, , G . , , ( ) ( ) . - N .

- G GA , A AE

-, G , Q [ ] ():

1. 0Q [ ]A , AG , (1)

2. 1Q[ ] , (2)

3. Q[ ] Q[ ] Q[ ]A B A B , (3) A, BG , A B = .

Q | G (1) GA :

AA . , 3 3, 1( )

Mm mG G , ,

1Mm mA G , ( 1 2m mG G ). (4)

1

Q[ ] Q[ ]M

mmA G

. (4)

(4) AA , (4) Q [ ]A .

1: (4) Q [ ]A , AA .

: AA - G : 1

Mm mA G 1

Jj jA F . m j m jC G F .

:

(1) Q | G , Q G . Q | G - Q | E , E G .

16/7/14 3:32 AM ..

70 . 2

() G , () , () 1 1

M Jm j m jC A ,

() 1Mm m j jC F 1

Jj m j mC G

Q | G ,

11 1 1 1

Q [ ] Q [ ] Q [ ]M M J M

Jm j m j m j

m m j m

G C C

11 1 1 1

Q [ ] Q [ ] Q [ ]J J M J

Mj m m j m j

j j m j

F C C

.

, 1 1Q[ ] Q[ ]

M J

m jm jG F

,

Q [ ]A () AA .

1: Q | A 1, 2, 3, (probability premeasure)(2).

( ) , , - A . - Q | A . Q [ ] , - () AA A .

1: 0,1[ ]Q A : , A () 1, 2, 3.

, :

) [ ]Q , ,

( )n nA A , nA , 1

nn

A A

A ,

1

[ ] [ ]nnA AQ Q

. (5)

) [ ]Q , , ( )n nB A ,

lim lim 0[ ] [ ] [ ]n nn nB BQ Q Q . (5)

(2) 1, Q| G .

C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc

71

: ) ): (5) 1

nn

A A

A nA A

. [ ]Q . ( )n nB

A . , ( )n nB :

1 2 1... ...n nB B B B , lim n n nn B B .

, 1n n nC B B , nC A , 1n nC B , n m n mC B , m . Q | A ( A , ), 1[ ] [ ] [ ]n n nnB C CQ Q Q , , [ ]nCQ . ,

0[ ] [ ] [ ]mm

n m n nn m

B C CQ Q Q

. ) ).

) ): [ ]Q , (5). , , Q | A . - ( )n nA A ,

n nA A A . 1mm n nB A A .

mB A .

[ ]Q , 0[ ]m mBQ , , mB Q | A ,

1 1

1 1

0 lim lim

lim ,

[ ] [ ] [ ]

[ ] [ ] [ ] [ ]

m m

n nm mn n

m

n nm

n n

A A A A

A A A A

Q Q Q

Q Q Q Q

,

. . : 1. ) ) - A . , n nA A . 2. ) ( ) A . 3.

16/7/14 3:32 AM ..

72 . 2

Q | A AA .

2: 0,1[ ]Q A : , A () 1, 2, 3.

, :

) [ ]Q (. (5)). ) [ ]Q A . , ( )n nA A , lim nn A A ,

AA lim lim[ ] [ ] [ ]n nnn A A AQ Q Q . (6)

: ) ): ( )n nA A , lim nn A A A . , ( )n nA A -

. ) lim 0[ ]nn A AQ . ,

lim lim lim[ ] [ ] [ ] [ ] [ ]n n nn nn A A AA A AQ Q Q Q Q (3).

(6).

( )n nA A , lim nn A A ,

AA . , ( )n nA A lim nn A A

A .

lim [ ]nn AQ lim[ ] [ ]nn A AQ Q . (#1) , lim lim lim 1 lim[ ] [ ] [ ] [ ] [ ]n n n nn n n nA A A AQ Q Q Q Q , lim lim lim 1 lim[ ] [ ] [ ] [ ]n n n nn n n nA A A AQ Q Q Q , 1[ ] [ ] [ ] [ ] [ ]A A A AQ Q Q Q Q . (#1), (6).

) ): . 2.

: 4. ) ) - A . , n nA A . 5. , Q | A , ) ) ).

(3) [ ] [ ] [ ]n nA AA AQ Q Q nA A .

C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC

BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc

73

, Q | G , - 1, 2, 3, Q |A , = GA , ( 1). Q |A -, , 1 2, , ... , MA A A A ( 2). - Q |A . , Q |A (, A ). ( ), Q |A n nA A . Q |A : Q |A .

3 [Hopfs extension Theorem/Lemma]: Q |A , , A

P , () 1, 2, 3.

, :

i) Q[ ] (. (5)).

ii) Q |A Q | C , C - A . , Q |A - C A . .

: 6. (. 1, (5)) i) - 3

( )

lim lim 0 .Q[ ]n n

n n n nn n

B

B B B

A

( ) i) :

i' ) ( )

lim 0 lim .Q[ ]n n

n n n nn n

B

B B B

A

7. , () - . - i' ).

16/7/14 3:32 AM ..

74 . 2

8. ( ) i' ), , ( / ). , ,X d - () , i i IG i i IF X , , fin IP I , , ,X d :

i i I i fin i M ii IG X X G M I X G P ,

i fin i M i i I ii IF X M I F F P . : ,X d (, i M iF M ) (, i I iF ). , , () ( ), - ( n nB ) i' ).

, 3 , , . , ( ) , - . ( ) P , 3 (Hopfs extension Theorem/Lemma) .

--------------------------------------------------

- (Caratheodory method)

2: *Q | P (outer or exterior measure), :

1. * 0Q [ ] , (7)

2. * *Q [ ] Q [ ]E F E F , (4) (7)

3. 1

* *Q [ ] Q [ ]m m mm

E E E E

, (5) (7)

(4) , *Q | P .


75

, , mE F E P .

: 1. *Q | P 0, , * 0,Q P: . 1 2. *Q [ ]H , H , 1,2,3, .

*Q | P ( ), *Q | P ( ).

2. 2 () 3, . ( ) (-). 3. ( ) - , . ., .., (Munroe, 1953), Sec. 11. ( 3, ), Q |A . 4. , , to 1918 (Constantin Carathodory)(6). . - , , , , - . A.N. Kolmogorov, - (1933). :

3: *Q | P . E

* * *Q [ ] Q [ ] Q [ ]W W W E W E P (8) *Q (7). (-

(5) , *Q | P [ ]. (6) Wikipedia http://en.wikipedia.org/wiki/Constantin_Carath%C3%A9odory. - (Georgiadou, 2004), Constantin Carathodory: Mathematics and Politics in Turbulent Times. (7) ( (8)) . ?. .., 127 (Hewitt & Stromberg, 1965) How Caratheodory came to think of this definition seems mysterious, since it is not in the least intuitive.

16/7/14 3:32 AM ..

76 . 2

(8)) E (divides well) . *Q

** ,Q U U . , . , ** ,Q U U .

4 [ ]: *Q | P ** ,Q U U *Q .

, : () *U , () * *Q | U () , () *U , () * *Q | U .

, * *Q | U .

: 5. , () () () (), () -. , , () () . -, ( ), ( ) . , . 6. *Q | P ( ), W W E W E , * * *Q [ ] Q [ ] Q [ ]W W E W E . , (8) () -

* * *Q [ ] Q [ ] Q [ ]W W W A W A P . (9)

4:

(): E W P , E W P (). , *U . ,A B

W P . A B . , * * *Q [ ] Q [ ] Q [ ]W W A W A [ A W ]

* * *Q [ ] Q [ ] Q [ ]A BW A B W AW (#1)



77

[ B W , W A ]. ,

.

A B A B

W A B A W A B W A

A B A

W A B

, W A B W A W A B , , - *Q | P ,

* * *Q [ ] Q [ ] Q [ ]W A B W A W A B . (#1),

* * *Q [ ] Q [ ] Q [ ]W W A B W A B . , A B W P . , , *U , .

(): 1 2 1 2 1 2* *, , ,A A A A A A A U U . -

1 2

* * *Q [ ] Q [ ] Q [ ]A A A . (#2) : 1

*A U , 1A - , W A , W . , 1 1

* * *Q [ ] Q [ ] Q [ ]W A W A A W A A . 1 1 1 2,A A A A A A . ,

1 2* * *Q [ ] Q [ ] Q [ ]W A W A W A . (#3)

(#3) W A (#2). (#3), : *,nA n U , nA , 1

*n NN n nA A

U ,

, W ,

1 1* * *Q [ ] Q [ ] Q [ ]

n Nn NN n n nn

W A W A W A

. (#4)

(): *,nA n U , 1 21 2 n nn n A A .

1*n p

p n nB A U limn n ppA A B .

*A U . *pB U , -

p ( *U , ()), pB W :

* * *Q [ ] Q [ ] Q [ ]p pW W B W B . (#5) , 1 1n p n pp n n n nW B W A W A . (#4),

16/7/14 3:32 AM ..

78 . 2

1 1* * *Q [ ] Q [ ] Q [ ]

n pn pp n n nn

W B W A W A

. (#6)

, pB A , p pB A W B W A , -

*Q | P , * *Q [ ] Q [ ]pW B W A . (#7) (#6) (#7), (#5)

1

* * *Q [ ] Q [ ] Q [ ]n p

nnW W A W A

.

p ,

1

* * *Q [ ] Q [ ] Q [ ]n

nnW W A W A

. (#8)

, n nA A , n nW A W A , n nW A W A . , 3,

1

* *Q [ ] Q [ ]n

nnW A W A

. (#9)

(#9) (#8) * * *Q [ ] Q [ ] Q [ ]W W A W A , () n nA A

W . , *U . , - (. ?? 5), .

(): (#8) W A , n nA A A ,

1

* *Q [ ] Q [ ]n

nnA A

.

*Q | P (.., W A (#9)). , , n nA A ,

*,nA n U , nA ,

1

* *Q [ ] Q [ ]n

nnA A

.

, * *Q | U *Q | P *U . 4.

( ) , .

5 [ ]: * *Q | U , () 4, :



79

(1) A , * 0Q [ ]A , *U . (2) *Q . , E A , * 0Q [ ]A , E *U . (3) 1 2A A A , 1 2

*,A A U , 1 2* *Q [ ] Q [ ]A A , -

*AU , 1 2* * *Q [ ] Q [ ] Q [ ]A A A .

: (1) A , * 0Q [ ]A , W :

* * *Q [ ] Q [ ] Q [ ]W W A W A . , W A A , , *Q [ ] , * * 0Q [ ] Q [ ]W A A . , * *Q [ ] Q [ ]W W A , , *Q [ ] . (2) E A * 0Q [ ]A , , *Q [ ] , - * * 0Q [ ] Q [ ]E A , * 0Q [ ]E . , *E U , (1).

(3) 1 2A A A , , 1 2 1 1 2 1A A A A A A A A . , 2 1A A *U 2 1 2 1

* * * 0Q [ ] Q [ ] Q [ ]A A A A . , 1 *A A U , - (2). , 1 1A A A A *U , - *AU . , 1 2 1A A A A ,

- *Q [ ] , 1 1 2 1

* * * *0 0Q [ ] Q [ ] Q [ ] Q [ ]A A A A A A , 1

* *Q [ ] Q [ ]A A . .

(2), , (, !) - () E . , Q |E , E . .

4: Q |E ( E ). E AE 0Q[ ]A (, , E ), Q |E .

, (- 5).

Q | A () A . (

16/7/14 3:32 AM ..

81

) . *Q [ ] 0,1 : * 10,Q P: . * 1Q [ ]E (10) * 1Q [ ] Q[ ]E E A .

: ( )m mA A () E m mE A . ( )m mA - ( ). , (10) :

E , *Q [ ]E infimum 1Q[ ]mm A

, ( )m mA A E .

*Q | P , 1,2,3 ( (7,,)).

(1) P A . , * *0 0Q [ ] Q [ ] Q [ ] .

(2) E F , F ( A ) E . , E F . infimum infi-mum , .

(3) 1

*Q [ ]mm E

, .

1

*Q [ ]mm E

.

*Q [ ]E 1

*Q [ ]mm E

, mE P mnA A : m n mnE A .

*Q [ ] , (10), 0 , ( )mn nA

mE

1

* *2

Q [ ] Q [ ]mn m mn

A E

. , ,( )mn m nA

E :

m n mnE A

. , , (10),

16/7/14 3:32 AM ..

82 . 2

,

1, 1 1 1

1 1

* * *

* * .2

Q [ ] Q [ ] Q [ ]

Q [ ] Q [ ]

mn mnm n m n

m mmm m

E A A

E E

0 , .

, Q |A , *Q | P , . ( 4), *U , *Q - , , *Q [ ] *U . * *Q | U Q |A :

* UA *Q[ ] Q [ ]A A A A .

2 3.

2. *A U . AA . A (9). () ( )m mF A W , () AA . , ( )m mF A A ( )m mF A A W A W A , , m mF A W A m mF A W A , , (10),

1

* [ ]mm

F A W AQ

1

* [ ]mm

F A W AQ

. (#1) Q |A

1 1 1

.

Q[ ] Q[ ] Q[ ]

Q[ ] Q[ ] Q[ ]

m m m

p p p

m m mm m m

F F A F A

F F A F A

p , (#1),

1 1 1

* *Q[ ] Q[ ] Q[ ] Q [ ] Q [ ]m m mm m m

F F A F A W A W A

. infimum ( )m mF A W , (9). , , - A *U . ,

* * G A U U ( *U ).



83

3. *Q[ ] Q [ ]A A A A . , , AA . , A P AA . , (10), *Q [ ] Q[ ]A A . , *Q[ ] Q [ ]A A , ( )m mA A A , m mA A , (#2)

1

Q [ ] Q [ ]mm

A A

. (9) (#3) , , (10),

* inf , ( )Q [ ] Q [ ] Q [ ]m m mmA A A A A A .

(#3) Q |A , , 1.

(#2) A , (#2) A : m mA A A . (), ( ) mA A , A . :

m mA A A . (#4) (#4) mA A , , . , , A A , : 1 1B A A , 2 2 1B A A B , 3 3 1 2B A A B B , , 11mm m n nB A A B . mB , :

( )m mB A , m mB A , m m m mB A A A A . (#5) , , A :

m mA B [ ].

, (#5), Q |A (. 1), , m mB A Q |A , -

1 1

Q [ ] Q [ ] Q [ ]m mm m

A B A

, (#3). 3 ,

(9) Q|A .

16/7/14 3:32 AM ..

84 . 2

Q[ ]A *Q [ ]A , AA .

3 (Hopfs Extension Theorem/Lemma) :

Q |A P , *Q | P , *U , , , * *Q | U - *Q | P ()

* Q Q A . * *Q | U Q |A . * *Q | U - Q |A . - * *Q | U *Q | U , ( *Q | P *U ) ( Q |A *U ). 1.

4. , *U - A , * * A U U . :

* A U A *U ?

, - , Card c , A *U :

*Card A Uc . (11)

. , 0,1 , A , 0,1a b , ,Q[ ]a b b a . (. 2 4 5) A ( Borel

0,1B ) c . , , Cantor C (10)

(10) Cantor : 0,1 1/3, 1 / 3, 2 / 3 . , 1/3, 2 21 / 3 , 2 / 3 2 2( 2 3 1) / 3 , ( 2 3 2 ) / 3 . , 1/3 mth-.



85

(Borel) (11), Card C c , (12). , 2 c c . , Cantor C , , A 0,1B , . , ,

C *U , (2) - 5 . , , , , A *U .

Q |A * *|Q U , . , *Q [ ]E , *E U A , ( (10)). , ,

* *Q | U ( Q |A ), - . - 7, ( 6).

*Q | Q |A A . , - 3.

Cantor 0,1 . (11) 0 , 1C B Cantor:

13 1

0

3 1 3 20, 1 ,

3 3

m

m m

k

mk

k kC

.

. http://en.wikipedia.org/wiki/Cantor_set , Mohsen Soltanifar (2006) A Different Description of A Family of Middle-a Cantor Sets On A sequence of cantor Fractals. (12) Q[ ]C . 0 0 , 1C , 1 0 , 1 1 / 3, 2 / 3C ,

2 2 2 22 1 1 / 3 , 2 / 3 7 / 3 , 8 / 3C C , ... , lim mm

C C

,

limQ[ ] Q[ ]mm

C C

. mC

1 1 1 / 3Q[ ]C , 2 12 22 1 / 3 1 1 / 3 2 / 3Q[ ] Q[ ]C C , ,

011 2 /3Q[ ]m

n mn

n nC . lim 0Q[ ]m

mC

.

16/7/14 3:32 AM ..

86 . 2

1. Q |A ( ) *Q | P , *U * *Q | U . () E ( A , A ), (-) *Q | Q |E E *Q | Q | A A .

Q | Q |A A , Q | Q |A E * *Q | Q |A U , * A A E U .

() Q | |QE E .

Q |A

* *Q | U

*Q | P

*Q | Q |E E

*Q | Q | A A

|Q E



87

- Q | Q |G G

Q | G , - G P , ()

Q | Q |G G . , G , () Q[ ] . , ( ), Q | Q |G G , , .

6: G (), P | G , Q | G - G . :

P Q P Q G G . (12)

: GD , : P[ ] Q[ ]D D D GD . G , .

D Dynkin. ,

D1) D , 1P[ ] Q[ ] . D2) ,A B A B B A D D . , ,A B D P[ ] Q[ ]A A P[ ] Q[ ]B B , P[ ] Q[ ]B A B A , , A B ,

P[ ] P[ ] P[ ]B A B A Q[ ] Q[ ] Q[ ]B A B A . D3) 1, nnn n nnA A A A

D D

. nA D ,

n , lim limP[ ] Q[ ] P[ ] Q[ ]n n n nn nA A A A

[ P[ ] Q[ ] ] lim limP[ ] Q[ ]n nn nA A ,

, 1n nA A , P[ ] Q[ ]n n n nA A . D G Dynkin. , Dynkin,

16/7/14 3:32 AM ..

88 . 2

Dynkin

G

G G D .

, G D , P[ ] Q[ ] G . , 6 , P[ ] - - G , P | P |G G . , -, * *Q | Q |A U ,

*Q | Q |A A . * * *Q | Q | A U , A *U , *Q [ ] -

* U A . - A *U . (-- ).

( ) / ( ) . 3 Q |A ( ) * *Q | U , *U - A A . ,

*U () A . , ,

*Q | A * *Q | U ( - A ) ?

( , , - ):

1: A *U , 2: *Q [ ]W , *W AU .

() - .


89

A () . ,n n nE A A A A , ,n n nE A A A A , ,n n nE A A A A A , ,n n nE A A A A A . , , A A , , A A . , - - , , - . 5, - A A ,

A A A A A A . (13)

( ),

7: A , Q |A (13) * *Q | U Q |A -

. , W , A A ,

W A * *Q [ ] Q [ ]W A . (14) *Q [ ]W W *Q | P (, W *Q ),

*Q [ ]A *A A U * *|Q U .

: , (10), , n , nk kA A W ,

k nkW A 1

* 1Q[ ] Q [ ]nkk

A Wn

. (#1) 0n k nkA A .

(13) , A Q|A . . (Munroe, 1953), Sec. 12.

16/7/14 3:32 AM ..

90 . 2

01 1

* * *Q [ ] Q [ ] Q [ ] Q[ ]n nk nk nkk

k k

A A A A

. (#2)

(#1) (#2)

01

* * 1Q [ ] Q[ ] Q [ ]n nkk

A A Wn

. (#3) 00 0n nA A . nkA , (#1), 00W A , ( )

00* *Q [ ] Q [ ]W A . (#4)

, n

00 0 00 0* *Q [ ] Q [ ]n nA A A A .

(#3),

00 0* * * 1Q [ ] Q [ ] Q [ ]nA A W n , n ,

00* *Q [ ] Q [ ]A W . , (#4),

00

* *Q [ ] Q [ ]A W . , 00 n k nkA A , nkA A ,

00*A A U .

. 7 , *Q [ ]W *W AU .

7: E , Q |E . *Q | P (14), Q |E . , W ( -), E E ,

W E E *Q [ ] Q[ ]W E . (15) 1E , 2E (15),

1 2 0[ ]Q E E . (15)

: 7, E E , (15). , E , E E . (15) : 1 2,W E W E 1 2W E E . ,

(14) *Q Q E ( ).


91

( )15

1 1 2*Q[ ] Q [ ] Q[ ]E W E E

.

, 1 2 1 1 2 1Q[ ] Q[ ]E E E E E E . 1 2 1Q[ ] Q[ ]E E E . , 1 2 1 1 2 1 1 2 0Q[ ] Q[ ] Q[ ] Q[ ]E E E E E E E E . ( 1E , 2E ), 2 1 0Q[ ]E E . , 1 2 1 2 2 1 1 2 2 1 0[ ]Q Q[( ) ( ) ] Q[ ] Q[ ]E E E E E E E E E E .

( !). - W , , E E , ( W ) = ( E ). :

5: Q |E , W . E E - (15) (measurable cover) W . 1E , 2E , W , (15), , .

, , Q E , , (15), -. , AE , , AE , A N E 0Q [ ]N . , (-) ,A BE (,

0[ ]Q A B ) . , (justified, well-posed)

Qdef

0[ ]QA B A B ( 10). , - QQ /E E E [ ]A , [ ] 0[ ]QA C E A C : . QQ /E E Q, , Q E - ( 10). ( ) - :

6: , , Q E , *Q | P Q |E . () AE 0Q[ ]A Q .

(15) .., -.

16/7/14 3:32 AM ..

92 . 2

() 0 *Q ( ).

Q ( *Q ) (Q) ( * * *(Q ) ). ,

0(Q) Q [ ]N N E : . (16) * * * * 0(Q ) Q [ ]N N : . (16)

( 7 5) ! , , :

E *U . Q |E -

W .

E *U 8, . Q |E - W (W E ) , .

8: E , Q |E , * *Q | U Q |E

. , - *U U ,

U E Z , E E * * * *(Q )Z U . (17) , *Z Q B E 0Q[ ]B .

: 7, *U U Q C , , C E

U C * *Q [ ] Q[ ] Q [ ]U C C . , * * * 0Q [ ] Q [ ] Q [ ]C U C U . Q - H E C U Q BE U H . U C H U H . (16) , C H E * * * 0Q [ ] Q [ ] Q [ ]U H H C U . , U H *Q , Q , BE , Q . E C H Z U H , -

(16) C H U H . : C U , C H U H . , C U H C U H , C H U H H U H .



93

. 8.

8 *U *(Q )E . ,

9: Q |E , *Q | P , Q |E , *(Q )

*Q .

** (Q )U E . (18)

: *E U ( ) * *(Q )U ( 5), * *(Q )E U . , * *(Q ) E U . - 8, (17).

8 9 . ,

*U E *(Q ) , , -, * *Q | U Q | A .

10: E , Q |E , *Q | P Q |E

.

* * 0[ ]QU E U E U E : . (19)

:

* 0[ ]QU E U E C E : (#1) (19). , *C U . , ( ) C .

C E . (, U E E U E , * 0[ ]Q U E ). C . i) , C (;). ii) U C . , U C , U E U E (17) * 0[ ]Q U E ,

(17) :

16/7/14 3:32 AM ..

94 . 2

E E . iii) ,nU n , C , C . , nU C , nE E ,

* 0[ ]Q n nU E . , n n n n n n nU E U E

( 7), , *Q | P , * * * 0[ ] [ ] [ ]Q Q Qn n n n n n n n nnU E U E U E . (#2) n nU C , (#1)

n nE E E . i), ii), iii), C . , C E .

*(Q ) . , *(Q )U , * * 0[ ] [ ]Q QU U E . , * *(Q ) (Q ) E C E C .

, C C , E E , * 0[ ]Q C E , C E C E E E * * 0[ ] [ ]Q QC E C E ( C E C E ).

, * *(Q ) (Q ) C E C E . , *(Q )C E .

, 9, , *C U . - .

U E U E E U U E E U

U E E U E U U E U E



95

- * *Q | U Q | A .

A B , A B A B B A ,

A B . ( ) * [ ]Q A B , * [ ]Q A B * [ ]Q B A , * [ ]Q A * [ ]Q B . , * [ ]Q A B * [ ]Q A , * [ ]Q B

* * *Q [ ] Q [ ] Q [ ]A B A B . (20) () A A B B B A B A , () . , , ,

: A B A C C B . (21)

:

.

A C C B A C C A C B B C

A

A C B

A B

C C A C B

B C C

, , A BA BB Ax . x C . C A B , A CB . , A BA CBx C , A BA B A C BAC BC C . (20)

* * *lim 0 lim[ ] [ ] [ ]Q Q Qn nn nE A A E

.

, n nA *lim 0[ ]Q nn E A , E .

, *U U , n nA A . , , - ( ) A .

11 [ (Approximation property)]: * *Q | U |Q A

16/7/14 3:32 AM ..

96 . 2

. , *U U , n nA A . ,

** lim 0[ ]Qn nn nU A U A U A . (22)

() :

** 0 [ ]QU A U A U A . (22)

(20)

* ** 0 [ ] [ ]Q QU A U A U A . (23)

: , . () . () () * *Q | U *Q | P . : (22). - V , ,

*0 [ ]QV A V A V A : , (18) (#1) . (

*U V ). , A V . V .

i) V ( 0 , A A ). ii) V . , V V , , 0 , A A

* [ ]Q V A . V A V A , V A V A ,

* [ ]Q nV A * [ ]Q V A , A A . , V -

(#1), V V . iii) V . , 0 nV V , n . ,nA A ,

,*

2[ ]Q n n nV A

.

, ,n n n n n n nV A V A - * [ ]Q

(18) *Q * [ ]Q V A ( ) * |Q P . V A ( ) (

*U ). * [ ]Q *U ( ) .



97

, ,,

* *

* 1 .22

[ ] [ ]

[ ]

Q Q

Q

n n n n n n n

n n nn n

V A V A

V A

(#2)

(#2) , ,n nA (-

) A . (21). 1 , ,, ,

n Nn n n n n nV A A

:

1 , , , 1 ,

, 1 , .

n N n Nn n n n n n n n n n n n

nn n n n n N n

V A V A A A

V A A

* [ ]Q ,

1 ,

, 1 ,

*

* *

[ ]

[ ] [ ].

Q

Q Q

n Nn n n n

nn n n n n N n

V A

V A A

(#3)

,

1 ,*n

n N nA U 1 ,lim nn N nN A

. (#4)

, 1 ,* [ ]Q nn N nA * [ ]Q

* *Q | U . , (#4) ,

1 , 1 ,* *lim lim 0[ ] [ ]Q Qn nn N n n N nN NA A

.

, / 2 0 , 0 0 ( )N N ,

0 1 ,*

2[ ]Q nn N nN N A

.

(#2) (#3),

01 ,* 2 2[ ]Qn N

n n n nV A .

, n nV , A A , (#1),

01 ,n Nn nA A .

i), ii), iii) , V . A V , A V V . , ,

*U V . *U U . , 10, E A

* 0[ ]Q U E . , , 0 , A A ,

* [ ]Q E A . (21) , ,U A E , * [ ]Q , * * * *[ ] [ ] [ ] [ ]Q Q Q QU A U E E A E A . , U V , , *U V .

16/7/14 3:32 AM ..

98 . 2

.

: (22). *Q | P , (10), , 0 , U , n nA A U ,

n nU A 1

*02

[ ]

99

-

( ) Q |A ( ) - * *Q | U , - *U . * A A E U ,

Q | Q |A A , Q | Q |A E * *Q | Q |A U . (. 1). A *U , ( )

*U A E Z , E A * * *(Q )Z U ,

Z B A 0Q[ ]B . . 8. , 4 3 - 0,1Q |B , ,Q[ ]a b b a , / Q | A ( Q | E , * A E U ): , Q | A ( Q | E )

, * *Q | U ( -) , , A ( E ) *U .

/ , (-) . , - Q | E (20), U (U E ) -, E Q | E , . 8, 9, 10, ( , ) . 1 Q | Q |E E , . () ( 1), , .

Q | E ,

* 0(Q , ) Q[ ]Z B Z B B E E : (24)

(20) , / Q| E . .

16/7/14 3:32 AM ..

100 . 2

* (Q , )U E Z U E Z E E E : . (25)

: 1. * (Q , )E Q E . () - Q | E -. E E , * (Q , )E , . 2. - * *(Q ) ** ,Q U U , , *U () *Q | P , Q | E ( 1). , , ( - ) , , ** ,Q U U

* *(Q ) . * (Q , )E E , (24) (25), () Q | E .

12 [ -]: E , Q | E , E (25). ,

U E Z E , Q[ ] Q[ ]U E . (26) , () E , () Q | E , () Q | E .

() E . i) E , . ii) U E . U E . U E Z Z B ( 0Q[ ]B ),

0Z B Z B 0Z B Z . ,

0 0 0U E Z E B Z E B Z E B E Z . , 0 0 0 1 1U E B E Z E B E Z E B E Z E Z , 1E E B E 1 0 0Z E Z Z B , 0Q[ ]B . -

, U E . iii) ( )n nU E . n nU E .



101

n n n nU E Z nE E * (Q , )nZ E . ,

n n n n n n n n nU E Z E Z E Z , n nE E E * (Q , )n nZ Z E . n nU E .

i), ii) iii), E .

() Q | E ( ) (26) Q[ ]U . , U E , 1 1 2 2U E Z E Z , iE E , i iZ B E , 0Q[ ]iB , 1, 2i .

1 2 1 1 2 2 2 2 2 2E E E Z E E Z E Z B () 2 1 1E E B . ,

1 2 1 2 2 1

1 2 2 1 2 1 0 .

Q[ ] Q[( ) ( )]Q[ ] Q[ ] Q[ ] Q[ ]

E E E E E E

E E E E B B

1 2Q[ ] Q[ ]E E , Q[ ]U ( (26)) .

Q | E . 1. U E Z E , 0Q[ ] Q[ ]U E . 2. E , 1Q[ ] Q[ ] . 3. Q | E . nU E , n . ,

n n nU E Z , nE E , Q[ ] Q[ ]n nU E . , n n n n n nU E Z , n nE E * (Q , )n nZ E , Q[ ] Q[ ]n n n nU E . ,

Q [ ] Q[ ] Q[ ] Q[ ]n n n n n nn nU E E U , , Q | E .

() , Q | E . W E 0Q[ ]W . X W E (, , 0Q[ ]X ). W E , W N Z , N E , Z B B E 0Q[ ]B . , 0Q[ ]W , 0Q[ ]N . X W , , X W N Z N B E . , 0Q[ ] Q[ ] Q[ ]N B N B . ,

* (Q , )X E , , X E .

16/7/14 3:32 AM ..

102 . 2

(Munroe, 1953), Chapter 2, (Yamasaki, 1985), Chapter 1, (Billingsley, 1995), Section 3, (Ash & Dolans-Dade, 2000), Sections 1.2 and 1.3, (Skorokhod, 2005), (Richardson, 2009), Chapter 2, (Dudley, 1989), Chapters 3 and 4. , . , - , , , - () * *Q | U , () |Q A , () A - . A *U , , * |Q A * *Q | U .

, Hopfs Extension Theorem (Lemma) , , Yamasaki (1985), Section 1, Richardson (2009), Section 2.4. ( ) Munroe (1953), Section 12, Dudley (1989), Section 3.3. Skorokhod (2005), ( - ) () , , (. 28).

N , , , .

1. G , Q | G 1, 2, 3. Q |A ( Q | G ), GA , ,A BA .

2. P |A , - , , A ,

, , ,n nB A B n A :

0P [ ] , (.1)

1P [ ] P [ ]A A , (.2)

1P [ ] P [ ]A B A B , (.3)

1P [ ] P [ ]n n n nA A , (.3)

A B (21)

(21) A B B A , .



103

P [ ] P [ ] P [ ]B A A B A B , (.4)

, P [ ] P [ ] P [ ]A B A A B , (.4)

A B

P [ ] P [ ] P [ ]A B A B , (.4)

P [ ] P [ ]B A B A , (.5)

( , , ,n nA B A B A )

P [ ] P [ ] P [ ]A B A B , (.5)

P [ ] P[ ]n n nnA A , (.5) Bonferroni ( , , ,n nA B A B A )

1 2 31

... 1P[ ] P[ ]N

N nn

A A A A A N

, (.6) ( , , ,n nA B A B A )

P [ ] P [ ] P [ ] P [ ]A B A B A B , (.7)

P [ ] P [ ] P [ ] P [ ]

P [ ] P [ ] P [ ] P [ ]A B A B C

A B B C C A A B C

(.7)

( ).

3. 0,1Q[ ] A : . , ( )n nA A n nA A ,

Q [ ] Q[ ]n n nnA A . , , 0,1Q[ ] A : .

4. Q | E , n nA E ( E , ). ( n nA Q [ ]n nA ):

liminf liminf limsup limsupQ[ ] Q [ ] Q [ ] Q[ ]n n n nn n n nA A A A

. (.8)

(.8) n nA ? : (.8) -

16/7/14 3:32 AM ..

104 . 2

( , -, 1 2). , , , - Q[ ] .

( )n nA E , limit inferior limit superior :

1 1

** liminf limsupn n n nn k n k k n kn

A A A A A A

. ()

k nn k

C A

, 1, 2, 3, ...k [] ()

k nn k

D A

, 1, 2, 3, ...k [] ()

( ()) , . ,

1 1*

lim lim[ ] [ ] [ ] [ ] [ ]kC

n k k kk kk n k kA A C C CQ Q Q Q Q

()

1 1

* lim lim[ ] [ ] [ ] [ ] [ ]kD

n k k kk kk n k kA A D D DQ Q Q Q Q

. ()

( )k kC E ( )k kD E

k k kC A D , ,

[ ] [ ] [ ]k k kC A DQ Q Q . ()

( )k kA E , [ ]k kAQ , . , , , limit inferior limit superior. liminf [ ]kk AQ

limsup [ ]kk

AQ

. , (), () (),

lim liminf liminf[ ] [ ] [ ]k k kk k kC C AQ Q Q ()

limsup limsup lim[ ] [ ] [ ]k k kkk kA D DQ Q Q

. ()

(), (), () (),

** lim liminf limsup lim[ ] [ ] [ ] [ ] [ ] [ ]k k k kk k kk

A C A A D AQ Q Q Q Q Q

.

( , lim inf lim sup). (.8).



105

, ( )k kA E A E , **A A A ,

**[ ] [ ] [ ]A A AQ Q Q . (.8)

liminf limsup[ ] [ ] [ ] [ ]k kk kA A A AQ Q Q Q

,

[ ]k kAQ lim lim[ ] [ ] [ ]k kk kA A AQ Q Q ,

Q[ ] (, -).

5. * * *(Q ) *Q ( ), .

6. * *Q | Q |A U - 7, (14).

7. n n n n n n nU E U E .

8. A B A B , A C C B A B B C C A A B C A B C .

9. * *Q | U Q | A , A B *U ,

* * *Q [ ] Q [ ] Q [ ]A B A B . . 10. , , Q E ,A B E . - 0Q A B Q E . , - QQ /E E , [ ]A ,

[ ] 0[ ]QA C E A C : , Q, , Q E .

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