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Let B n
B . Then x
(
B n
)
x
(
B
)
, hence A . x
(
B n
)
A . x
(
B
)
. Therefore
ν (
B n
)=
m
(
A . x
(
B n
))
m
(
A . x
(
B
)) = ν (
B
)
.
Theorem 5.2.
Let x :
σ ( J ) →F
be an observable, m :
F→ [
0, .1
]
be a state, and let A
∈F
.
f 1
Then there exists a Borel measurable function f : R
R (i. e. B
σ ( J )=
(
B
) σ ( J )
)
such that
(
(
)) =
m
A . x
B
fdm x
B
∈ σ ( J )
for any B
.If g is another such a function, then
( {
(
) =
(
) } )=
m x
u
R ; f
x
g
x
0.
μ
ν
σ ( J ) [
]
Proof. Define
,
:
0, 1
by the formulas
μ (
B
)=
m x (
B
)=
m
(
x
(
B
))
,
ν (
B
)=
m
(
A . x
(
B
))
.
Then
μ
,
ν
:
σ ( J ) [
0, 1
]
are measures, and
ν μ
.
By the Radon - Nikodym theorem there exists exactly one function f : R
R (with respect to
the equality
μ
- almost everywhere) such that
(
(
)) = ν (
)=
μ =
∈ σ ( J )
m
A . x
B
B
fd
fdm x , B
.
B
B
Definition 5.1.
σ ( J ) →F
∈F
Let x :
be an observable A
. Then the conditional probability
f 1
(
|
)=
∈J =
(
) ∈ σ ( J ))
p
A
x
f is a Borel measurable function (i. e. B
B
such that
p
(
A
|
x
)
dm x
=
m
(
A . x
(
B
))
B
∈ σ ( J )
for any B
.
7. Algebraic world
At the end of our communication we shall present two ideas.
The first one is in some
algebraizations of the product
=( μ A .
ν A + ν B − ν A .
ν B )
A . B
μ B ,
.
The second idea is a presentation of a dual notion to the notion of IF -event.
In MV-algebras the product was introduced independently in [56] and [47]. Let us return to
Definition 1.3 and Example 1.5.
Definition 6.1.
, where M is an MV-algebra, and .
is a commutative and associative binary operation on M satisfying the following conditions:
An MV-algebra with product is a pair
(
M ,.
)
(i) 1. a
=
a
(
¬
)=(
) ¬ (
)
(ii) a .
b
c
a . b
a . c
.
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