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In the discrete case we have
k
1 = i x i p i ,
x 2
E
(
)=
in the continuous case we obtain
x 2
t 2
E
(
)=
ϕ (
t
)
dt .
5. Sequences
In the section we want to present a method for studying of limit properties of some
sequences
(
)
B (
) →F
of observables ([7], [25], [31], [32], [49]). The main idea
is a representation of the given sequence by a sequence of random variables
x n
n , x n :
R
( ξ n
)
n ,
ξ n :
( Ω
,
S
, P
)
R . Of course, the space
( Ω
,
S )
depends on a concrete sequence
(
x n
)
n , for different
sequences various spaces
( Ω
,
S
, P
)
can be obtained.
The main instrument is the Kolmogorov consistency theorem ([67]). It starts with a sequence
of probability measures
( μ n
)
n ,
μ n :
σ ( J
) [
0, 1
]
such that
n
μ n + 1 | σ ( J n ) ×
R
= μ n
i. e.
μ n + 1 (
A
×
R
)= μ n
(
A
)
for any A
σ ( J
)
(consistency condition). Let
C
be the family of
n
all cylinders in the space R N , i. e. such sets A
R N that
A
= { (
t n
)
n ;
(
t 1 , ..., t k )
B
}
,
R k
k
∈B (
)= σ ( J
)
where k
N , B
. Then by the Kolmogorov consistency theorem there exists
exactly one probability measure
P :
σ ( C ) [
0, 1
]
such that
(
)= μ k (
)
P
A
B
.
(6)
π n : R N
R n ,
If we denote by
π n the projection
t i ) i = 1 )=(
((
)
π n
t 1 , t 2 , ..., t n
,
then we can formulate the assertion (6) by the equality
( π 1
P
(
B
)) = μ n (
B
)
,
(7)
n
for any B
∈C
.
Theorem 4.1.
Let m be a state on a space
F
of all IF-events.
Let
(
x n
)
n be a sequence of
R n
observables, x n :
B (
R
) →F
, and let h n :
B (
) →F
be the joint observable of x 1 , ..., x n , n
=
R n
1, 2, .... If we define
μ n :
B (
) [
0, 1
]
by the equality
μ n
=
m
h n ,
then
( μ n
)
n satisfies the consistency condition
| ( σ ( J
) ×
)= μ n .
μ n + 1
R
n
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