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In-Depth Information
A
C
Notice, that it is
m
ʻ
(
)
=
N
ʻ
(
m
ʻ
(
))
, with the Sugeno's negation
N
ʻ
(
)
=
A
x
1
−
x
x
(ʻ >
−
)
1
.
1
+
ʻ
Remarks 7.4.3
•
With the t-conorm
S
ʻ
(
x
,
y
)
=
x
+
y
+
ʻ
xy
,itfollows
m
ʻ
(
A
∪
B
)
=
S
ʻ
(
m
ʻ
(
A
),
m
ʻ
(
B
))
,if
A
∩
B
= ∅
.
•
When
ʻ
=
0,
m
0
is just a probability defined in
P
(
X
)
since it results
m
ʻ
(
A
∪
B
)
=
m
ʻ
(
A
)
+
m
ʻ
(
B
)
when
A
∩
B
= ∅
, that is,
m
0
is an additive measure. In addition,
A
C
since
N
0
(
x
)
=
1
−
x
,itis
m
0
(
)
=
1
−
m
0
(
A
)
.
•
Notice that the axioms required for a
ʻ
-measure do not individuate a single one
of them. For example, with
ʻ
=
0 what is obtained is the set of all probabilities
on
P
(
X
)
.
•
If
ʻ
∈
(
−
1
,
0
)
, it results
A
∩
B
= ∅ :
m
ʻ
(
A
∪
B
)
m
ʻ
(
A
)
+
m
ʻ
(
B
),
that is,
if
ʻ
∈
(
−
1
,
0
)
,
all the corresponding
ʻ
-
measures are sub-additive
.Asit
is easy to prove, if
ʻ
∈
(
0
,
+∞
)
,
m
is super-additive.
ʻ
•
As it is well known, if
X
is a finite set
X
={
x
1
,...,
x
n
}
, all probabilities
m
0
:
P
(
X
)
ₒ[
0
,
1
]
are defined by choosing n numbers
m
0
(
{
x
i
}
)
∈[
0
,
1
]
,1
i
n
,
verifying
i
=
1
m
0
(
{
x
i
}
)
=
1, because 1
=
m
0
(
{
x
1
,...,
x
n
}
)
=
m
0
(
x
1
)
+
...
+
m
0
(
x
n
)
. Something analogous happens with
ʻ
-measures when
X
={
x
1
,...,
x
n
}
.
For example, if
X
={
x
1
,
x
2
,
x
3
}
, it follows
=
m
ʻ
(
)
=
m
ʻ
(
{
x
1
,
x
2
,
x
3
}
)
1
X
=
m
ʻ
(
{
x
1
,
x
2
}∪{
x
3
}
)
=
m
ʻ
(
{
x
1
,
x
2
}
)
+
m
ʻ
(
x
3
)
+
ʻ
m
ʻ
(
{
x
1
,
x
2
}
)
m
ʻ
(
x
3
)
3
2
m
ʻ
(
=
m
ʻ
(
x
i
)
+
ʻ
m
ʻ
(
x
i
)
m
ʻ
(
x
j
)
+
ʻ
x
1
)
m
ʻ
(
x
2
)
m
ʻ
(
x
3
).
i
=
1
1
=
i
<
j
=
3
ʻ
∈
(
−
,
+∞
)
, the values
m
ʻ
(
x
i
)
and for each
1
,1
i
3, are to be taken veri-
=
i
=
1
m
0
(
ʻ
=
x
i
)
fying this equation that, of course, with
0 reduces to 1
.
In the case that, for example, is
m
(
x
1
)
=
0, follows
1
=
m
ʻ
(
x
2
)
+
m
ʻ
(
x
3
)
+
ʻ
[
m
ʻ
(
x
2
)
m
ʻ
(
x
3
)
]=
m
ʻ
(
x
2
)
+
m
ʻ
(
x
3
)
+
ʻ
m
ʻ
(
x
2
)
m
ʻ
(
x
3
)
1
−
m
ʻ
(
x
2
)
0
.
3
that is:
m
ʻ
(
x
3
)
=
. With
m
ʻ
(
x
2
)
=
0
.
7, results
m
ʻ
(
x
3
)
=
. With
1
+
ʻ
m
ʻ
(
x
2
)
1
+
0
.
3
ʻ
0
.
3
ʻ
=
1:
m
1
(
x
3
)
=
03
=
0
.
29. That is: ameasure
m
1
is defined in
X
={
x
1
,
x
2
,
x
3
}
,
1
.
by
m
1
(
x
1
)
=
0,
m
1
(
x
2
)
=
0
.
7,
m
1
(
x
3
)
=
0
.
29. Notice that, since
m
1
is not a
probability, it is
i
=
1
m
(
x
i
)
=
0
.
99
<
1.
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