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 .
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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)
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC
BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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, :
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC
BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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 ..
80 . 2
), A () , ( ) .
A *U ( ) . .
-------------------------------------------
- Q |A * *Q | U
, . 3, - . i) ii). 3 , . 1 Q |A *Q | P , P , , * *Q | U , *U , *Q . 2 A *U . 3 *Q[ ] Q [ ]A A AA . ,
* *Q | U Q |A , *U , A . Q |A 3 . 4 ,
*U A . ( ). ( 6), -. ( 3): ii) i): ) ) 1.
i) ii): 1. Q |A . E ,
1
* inf ,Q [ ] Q[ ]m m m m mm
E A E A A
A
: . (8) (10)
*Q [ ] E . , ( ),
1Q[ ]mm A
( ), infimum - (
(8) mA .
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC
BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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 ).
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC
BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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-.
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC
BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC
BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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 .
() - .
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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 ( ).
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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 .
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC
BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC
BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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 ( ) .
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC
BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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 .
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC
BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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 , .
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC
BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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).
C:\Users\mathan\Dropbox\MA8HMATA\BOOK_Math_20_Stoch_book_B\Chapter_02a_MEASURE-THEORETIC
BACKGROUND\08_Abstract Construction of Probability Measures-NEW.doc
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 .
16/7/14 3:32 AM ..