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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