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(, F, )

G F

f L1(, F, )

g L1(, G, )

A

f d=

A

g d

A G E[f | G] g

f

G

0 G 0 (, G)

0(A) =(A) A G (, G)

(A) =A

f d0 A G 0

g L1(, G, )

A

g d0= (A)

A G

A

g d =

A

g d0= (A) =

A

f d0=

A

f d

E[f | G]

G

A

f d=A

E[f | G]d

A G

E[f | G] E[f | G]

E[f | G] f

L1(, F, )

G = {, } E[f | G] =

f d := E[f] E[f]

G

A

E[f] d= A

f d

A=

A=

G=F E[f | G] =f f G

P F

G=(P)

E[f | G]() =h() :=

CP

1C() 1

(C)

C

f d

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h G A G

(P) ={ CQC : Q P}

A= CQ

C Q P

A

h d=

CQ

C

h d=

CQ

C

( 1

(C)

C

f d) d=

CQ

C

f d=

A

f d

G F E[ | G]

L1(, F, ) L1(, G, )

E[ | G]

E[ | G]

g L(, G, ) E[g f | G] =g E[f | G] fL1(, F, )

B G fL1(, F, )

E[E[f | G]| B] =E[E[f | B]| G] =E[f | B]

f, g L1(, F, ) , R E[f | G] + E[g | G]

G

A G

A

E[f | G] + E[g| G] d= A

E[f | G] d + A

E[g| G] d=

=

A

f d +

A

g d=

A

f+ g d

E[f+ g | G] =E[f | G] + E[g| G]

fL1(, F, ) E[ | G] E[f | G]L1

fL1 A = {E[f | G] 0} E[f | G] G

A, Ac G

E[f | G]L1 =

A

E[f | G] d +

Ac

E[f | G] d=

=

A

f d +

Ac

f d

A

|f| d +

Ac

|f| d= fL1

f L1(, F, ) A G 1A E[f | G] G

B G

B

1A E[f | G] d=

AB

E[f | G] d=

AB

f d=

B

1A f d=

B

E[1A f | G] d

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E[ | G]

g L(, G, )

g 0

{sj}j=1

L(, G, ) 0 s1 s2

... g sjj

g B G

B

g E[f | G] d= limj

B

sj E[f | G] d= limj

B

sj f d=

B

g f d

E[g f | G] = g E[f | G] g g = g+ g

E[f | B]

G

E[E[f | B]| G] =E[f | B]

E[E[f | G]| B] =E[f | B] B B

B

E[f | G] d=

B

f d=

B

E[f | B] d

E[f | B] B E[E[f | G]| B] =E[f | B]