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hence
h
(
x
(
.
y
(
×
)=
)
)
C
D
C
D
.
Analogously
h
(
x
(
.
y
(
C
×
D
)=
C
)
D
)
.
If we define
h
(
h
(
2
(
)=(
)
−
))
∈
σ
(
J
)
h
A
A
,1
A
,
A
,
then
x
(
.
y
(
x
(
.
y
(
(
×
)=(
)
)
−
)
)) =
(
)
(
)
.
Now we shall present two applications of the notion of the joint observable. The first is the
definition of function of a finite sequence of observables, e.g. their sum. In the classical case
h
C
D
C
D
,1
C
D
x
C
.
y
D
ξ
+
η
=
g
◦
T
:
Ω
→
R
where
g
(
u
,
v
)=
u
+
v
,
T
(
ω
)=(
ξ
(
ω
)
,
η
(
ω
))
. Hence
ξ
+
η
can be defined by the help of
pre-images:
(
ξ
+
η
)
−
1
T
−
1
g
−
1
=
◦
:
B
(
R
)
→S
.
Definition 3.4.
B
(
)
→F
→
Let
x
1
, ...,
x
n
:
R
be observables,
g
:
R
n
R
be a measurable function.
Then we define
(
)
B
(
)
→F
g
x
1
, ...,
x
n
:
R
by the formula
g
−
1
g
(
x
1
, ...,
x
n
)(
C
)=
h
(
(
C
))
,
C
∈B
(
R
)
,
R
n
B
(
)
→F
where
h
:
is the joint observable of the observables
x
1
, ...,
x
n
.
+
+
B
(
)
→F
(
+
+
Example 3.1.
x
1
...
x
n
:
R
is the observable defined by the formula
x
1
...
g
−
1
R
n
is the joint observable of
x
1
, ...,
x
n
, and
g
:
R
n
)(
)=
(
(
))
B
(
)
→F
→
x
n
C
h
C
, where
h
:
R
is defined by the equality
g
u
n
.
The second application of the joint observable is in the formulation of the independency.
(
u
1
, ...,
u
n
)=
u
1
+
...
+
)
n
=
1
Definition 3.5.
Let
m
:
F→
[
0, 1
]
be a state,
(
x
n
be a sequence of observables,
n
)
n
=
1
σ
(
J
)
→F
(
=
)
(
h
n
:
be the joint observable of
x
1
, ...,
x
n
n
1, 2, ...
.
Then
x
n
is called
independent, if
(
(
×
×
×
)) =
(
(
))
(
(
))
(
(
))
m
h
n
C
1
C
2
...
C
n
m
x
1
C
1
.
m
x
2
C
2
.....
m
x
n
C
n
for any
n
∈
N
and any
C
1
, ...,
C
n
∈
σ
(
J
)
.
Now let us return to the notion of mean value of an observable. In the classical case
(
◦
ξ
)=
◦
ξ
=
E
g
g
dP
gdF
Ω
R
where
F
is the distribution function of
ξ
.
Definition 3.6.
Let
x
:
B
(
R
)
→F
be an observable,
m
:
F→
[
0, 1
]
be a state,
g
:
R
→
R
be
a measurable function,
F
be the distribution function of
x
(
F
(
t
)=
m
(
x
((
−
∞
,
t
))))
.Thenwe
