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be an elementary cylindrical set of
A
1
.Denote
V
k,n
,k≥ n
, a set of all vectors
∈ X
k
,where
(
z
1
, ..., z
k
)
x
i
1
, ...x
i
n
is the sequence of all elements of
X
0
.Itisclear
V
k,n
× X
∞
is a cylindrical set of
that
A
.
× X
0
,x
i
j
∈ X
0
,j
Lemma 3.
If
C
n
=(
x
i
1
, ...x
i
n
)
=1
, ..., n
,
is an elementary
cylindrical set of
A
1
,
then
=
k≥n
{V
k,n
⊗ X
∞
}∩ X.
C
n
⊗ X
1
X
C
n
⊗ X
1
⊆
γ ∈ C
n
⊗ X
1
,γ→ α, β, α ∈ X
0
,β∈
Proof.
By definition
.Let
X
1
}
x
i
1
, ...x
i
n
,α
,where
α
is a sequence of
X
0
,and
α
is represented as
α
=
.
Let (
z
1
, ..., z
k
)
,k≥ n
, are the first
k
elements of
γ
,where
x
i
1
, ...x
i
n
is the
γ ∈ V
k,n
× X
∞
.
sequence of all elements of
X
0
.Then(
z
1
, ..., z
k
)
∈ V
k,n
,and
That is why
{V
k,n
⊗ X
∞
}∩ X.
γ ∈
k≥n
This is proof of
{V
k,n
⊗ X
∞
}∩ X.
C
n
⊗ X
1
⊆
k≥n
γ ∈ X
Prove the inverse implication of events. If
and
{V
k,n
⊗ X
∞
}∩ X,
γ ∈
k≥n
then there is a set
V
k,n
, that the first
k
elements (
z
1
, ..., z
k
)of
γ
are in
V
k,n
and
γ ∈ X
all elements of
X
0
in (
z
1
, ..., z
k
) are from the vector
x
i
1
, ...x
i
n
.If
,then
γ → α, β, α ∈ X
0
,β∈ X
1
and the first
n
elements of
α
form the vector
γ ∈ C
n
⊗ X
1
x
i
1
, ...x
i
n
.Then
α ∈ C
n
,and
.Hereweproved
{V
k,n
⊗ X
∞
}∩ X,
C
n
⊗ X
1
⊇
k≥n
and lemma is proved.
Corollary 1.
C
n
⊗ X
1
A ∩ X, A ∈A.
=
V
k,n
⊗ X
∞
is a cylindrical set of
Proof.
As
A
then
=
k≥n
{V
k,n
⊗ X
∞
}∈A.
A
Corollary 2.
∀C ∈A
1
,C⊗ X
1
∈.
Theorem 1.
Function
F
is
(
, A
1
)-
measurable
.
Proof.
The proof follows from lemmas 2 and 3 and corollary 2.
