elf3(2).pdf

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COLLOQUIUM MATHEMATICUM
T
1
t
L 2
t
1
·
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82
α
α
0 < α < 2
α = 2
α
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83
k≥3
{1, . . . , k}
[0, 1)
T
(X,B, )
T L 2 (X,B, )
U T (f ) = f◦T
T (f )
f◦T
C(T )
T
(X,B, )
T
U (L 2 (X,B, ))
T
σ A⊂B
T
T
(X/A,A, | A )
T| A
A
{T n
T
: n∈Z}
C(T )
(q n ) T q n →Id
T
S
(Y,C, ν)
T S
T×S
̺ (X×Y,B⊗C)
̺ X
̺ Y
̺ X = ̺| X = ̺ Y = ̺| Y = ν
T
S
J(T, S)
T×S
(X×Y,B⊗C, ̺)
(T×S, ̺)
J e (T, S)
̺
f⊗g d̺ =
Φ ̺ (f )g dν
X×Y
Y
J(T, S)
L 2 (X,B, ) L 2 (Y,C, ν)
J(T, S)
U T
U S
Φ : L 2 (X,B, )→L 2 (Y,C, ν)
Φ(1 X ) = 1 Y
Φ ∗ (1 Y ) = 1 X
Φ = Φ ̺ ̺(A×B) =
B Φ(1 A ) dν
A∈B B∈C
B(L 2 (X,B, ), L 2 (Y,C, ν))
J(T, S)
1244565457.001.png
 
84
J(T, S)
J(T, S)
J(T, S)
⊗ν Π X,Y (f ) =
X f d
R (Z,D, η)
Φ ̺ ∈J(T, S) Φ κ ∈J(S, R)
Φ κ ◦Φ ̺ ∈J(T, R)
T R
κ◦̺
J 2 (T ) J 2 (T )
S = T
J(T, T )
J e (T, T )
W∈C(T )
W (A×B)
= (A∩W −1 B)
T
W ∈J 2 (T ) W = T n
T
T
⊗
Id
X
T
A⊂B
λ
A
λ(A×B) =
E(1 A |A)E(1 B |A) d(| A )
X/A
A 1 ⊂A
T| A
A 1
T
A 1
T S
J e (T, S) ={⊗ν}
J(T, S) ={⊗ν}
R ⊥
T S
R
R
R ⊥
R ⊥
R ⊥
R ⊥
M(R ⊥ )
T
(X,B, )
̺∈J 2 (T )
̺◦̺∈J 2 (T ) (T×T, ̺)
B⊗{∅, X} {∅, X}⊗B
̺∈J 2 (T )
T
(T×T, ̺)
(T×T, ̺)
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85
T
(X,B, )
N 0 ⊂N
f, g∈L 2 (X,B, )
N\N 0
f◦T n , g→f, 11, g
f, g, h∈L ∞ (X,B, )
n→∞ n∈N 0
̺∈J(T )
f (T n x)g(x)h(T n y) d̺(x, y)→
f (x)h(y) d̺(x, y)
g(x) d(x).
X×X
X
X
f (T n x)g(x)h(T n y) d̺(x, y) =
f (T n x)g(x)Φ ∗ ̺ (h◦T n )(x) d(x)
X×X
X
(fΦ ∗ ̺ (h))◦T n g d
=
X
fΦ ∗ ̺ (h) d
→
g d =
f⊗h d̺
g d.
X
X
X×X
X
T S
σ
(X,B, ) (Y,C, ν)
T S
σ T×S
̺ (X×Y,B⊗C)
̺ X ̺ Y
̺ X ≤ ̺ Y ≤ν
f (x)g(y) d̺(x, y) =
V (f )g dν,
X×Y
Y
V : L 2 (X,B, )→L 2 (Y,C, ν)
U T
U S
V : L 2 (X,B, )→L 2 (Y,C, ν)
f∈L 2 (X,B, )
V f≤1
V ∗ g≤1
g∈L 2 (Y,C, ν) 0≤g≤1
0≤f≤1
T = S
T n
X T n (1 A ) d = (A)
A⊂X
1 X
̺
̺ X ≤
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