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=
((
μ
A
+
μ
B
,
ν
A
+
ν
B
−
1
)
−
(
1, 0
))
∨
(
0, 1
)
=
=(
μ
A
+
μ
B
−
ν
A
+
ν
B
−
−
+
)
∨
(
)
1,
1
0
1
0, 1
=
((
μ
A
+
μ
B
−
)
∨
(
ν
A
+
ν
B
)
∧
)
=
1
0,
1
,
¬
=(
)
−
(
μ
A
,
ν
A
)=
A
1, 0
=(
1
−
μ
A
,0
−
ν
A
+
1
)=
=(
1
−
μ
A
,1
−
ν
A
)
.
Connections with the family of IF-sets (Definition 1.1) is evident. Hence we can formulate the
main result of the section.
S
)
F
=(
μ
A
,
)
Theorem 1.1.
Let (
Ω
,
be a measurable space,
the family of all IF-sets
A
ν
A
be
S
M
F⊂M
such that
μ
A
,
ν
A
are
-measurable. Then there exists an MV-algebra
such that
, the
operations
⊕
,
are extensions of operations on
F
and the ordering
≤
is an extension of the
ordering in
F
.
Proof. Consider MV-algebra
M
constructed in Example 1.5. If
A
,
B
∈F
, then the operations
on
M
coincide with the operations on
F
. The ordering
≤
is the same.
Theorem 1.1 enables us in the space of IF-sets to use some results of the well developed
probability theory on MV-algebras ([66 - 68]). Of course, some methods of the theory can be
generalized in so-called D-posets ([28]). The system
(
D
≤
,
−
,0,1
)
is called D-poset, if
(
D
,
≤
)
is partially ordered set with the smallest element 0 and the largest element 1,
−
is a partially
binary operation satisfying the following statements:
−
≤
1.
b
a
is defined if and only if
a
b
.
≤
−
≤
−
(
−
)=
2.
a
b
implies
b
a
b
and
b
b
a
a
.
≤
≤
−
≤
−
(
−
)
−
(
−
)=
−
3.
a
b
c
implies
c
b
c
a
and
c
a
c
b
b
a
.
3. Probability on IF-events
In IF-events theory an original terminology is used. The main notion is the notion of a state
([21], [22],[57], [58], [61][, [62]). It is an analogue of the notion of probability in the Kolmogorov
classical theory.
As before
F
is the family of all IF-sets
A
=(
μ
A
,
ν
A
)
such that
μ
A
,
ν
A
:
(
Ω
,
S
)
→
[
0, 1
]
are
S
-measurable.
Definition 2.1.
F→
[
]
A mapping
m
:
0, 1
is called a state if the following properties are
satisfied:
(
Ω
)=
(
Ω
)=
(i)
m
1
,0
1,
m
0
,1
0,
Ω
Ω
(ii)
A
B
=(
0
,1
Ω
)=
⇒
m
((
A
⊕
B
)) =
m
(
A
)+
m
(
B
)
,
Ω
(iii)
A
n
A
=
⇒
m
(
A
n
)
m
(
A
)
.
Of course, also the notion with the name probability has been introduced in IF-events theory.
Definition 2.2.
J
J
=
{
[
]
∈
Let
be the family of all compact intervals in the real line,
a
,
b
;
a
,
b
≤
}
F→J
R
,
a
b
. Probability is a mapping
P
:
satisfying the following conditions:
