Information Technology Reference
In-Depth Information
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
.
