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hence lim
n
→
∞
F
(
u
n
)=
0 for any
u
n
−
∞
.
R
2
. We have
=(
ξ
η
)
Ω
→
Of course, we must describe also the random vector
T
,
:
T
−
1
)=
ξ
−
1
)
∩
η
−
1
(
C
×
D
(
C
(
D
)
.
In the IF case we shall use product of functions instead of intersection of sets ([47], [56], [68]).
Definition 3.2.
The product
A
.
B
of two IF-events
A
=(
μ
A
,
ν
A
)
,
B
=(
μ
B
,
ν
B
)
is the IF set
A
.
B
=(
μ
A
.
μ
B
,1
−
(
1
−
ν
A
)
.
(
1
−
ν
B
)) = (
μ
A
.
μ
B
,
ν
A
+
ν
B
−
ν
A
.
ν
B
)
.
σ
(
J
)
→F
Definition 3.3.
Let
x
1
, ...,
x
n
:
be observables. By the joint observable of
x
1
, ...,
x
n
n
n
being the set of all intervals of
R
n
) satisfying the
we consider a mapping
h
:
σ
(
J
)
→F
(
J
following conditions:
R
n
(i)
h
(
)=(
1, 0
)
∩
=
∅
=
⇒
(
)
(
)=(
)
(
∪
)=
(
)
⊕
(
)
(ii)
A
B
h
A
h
B
0, 1
, and
h
A
B
h
A
h
B
,
=
⇒
(
(
)
(iii)
A
n
A
h
A
n
h
A
,
(
C
1
×
×
×
)=
x
1
(
C
1
)
(
)
(
)
∈J
(iv)
h
C
2
...
C
n
.
x
2
C
2
.....
x
n
C
n
, for any
C
1
,
C
2
, ...,
C
n
.
Theorem 3.1.
([63]) For any observables
x
1
, ...,
x
n
:
σ
(
J
)
→F
there exists their joint
n
observable
h
:
σ
(
J
)
→F
.
Proof.
We shall prove it for
n
=
2.
Consider two observables
x
,
y
:
σ
(
J
)
→F
.
Since
(
)
∈F
x
A
, we shall write
x
(
x
(
x
(
A
)=(
A
)
,1
−
A
))
and similarly
y
(
y
(
y
(
B
)=(
B
)
,1
−
B
))
.
By the definition of product we obtain
x
(
.
y
(
x
(
.
y
(
(
)
(
)=(
)
)
−
)
))
x
C
.
y
D
C
D
,1
C
D
.
Therefore, we shall construct similarly
h
(
h
(
(
)=(
)
−
))
h
K
K
,1
K
Fix
ω
∈
Ω
and define
μ
,
ν
:
σ
(
J
)
→
[
0, 1
]
by
x
(
y
(
μ
(
)=
)(
ω
)
ν
(
)=
)(
ω
)
A
A
,
B
B
.
μ × ν
Let
be the product of the probability measures
μ
,
ν
. Put
h
(
K
)(
ω
)=
μ
×
ν
(
K
)
.
Then
h
(
x
(
.
y
(
C
×
D
)(
ω
)=
μ
(
C
)
.
ν
(
D
)=
C
)
D
)(
ω
)
