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7.3 Types of Measures
, where m is a fuzzy measure, if for some negation
(
Given a triplet
(
X
,
F
,
m
)
N
)
and
some union
+
(
S
)
,is
μ
˃
, then
m
1. When
(μ
+
˃)
m
(μ)
+
m
(˃)
,
m
is sub-additive
μ
˃
, then
m
2. When
(μ
+
˃)
m
(μ)
+
m
(˃)
,
m
is super-additive,
and when
m
is both sub-additive and super-additive, that is
μ
˃
,
When
then
m
(μ
+
˃)
=
m
(μ)
+
m
(˃)
,
m
is additive.
This classification (once completed with those measures that are neither sub-
additive, nor super-additive), in the case in which
X
, particularizes
μ, ˃
∈{
0
,
1
}
to:
•
If
A
∩
B
= ∅
, and
m
(
A
∪
B
)
m
(
A
)
+
m
(
B
)
, m is sub-additive
•
If
A
∩
B
= ∅
, and
m
(
A
∪
B
)
m
(
A
)
+
m
(
B
)
, m is super-additive
•
If
A
∩
B
= ∅
, and
m
(
A
∪
B
)
=
m
(
A
)
+
m
(
B
)
, m is additive.
|
A
|
Example 7.3.1
The measure
m
(
A
)
=
, in a finite set
X
={
x
1
,...,
x
n
}
, is addi-
n
tive.
7.4
λ
-Measures
With
F = P
(
X
)
,
m
ʻ
: P
(
X
)
ₒ[
0
,
1
]
is called a Sugeno's
ʻ
-
measure
if, with
ʻ >
−
1, it is:
1.
m
ʻ
(
∅
)
=
0
2.
m
ʻ
(
X
)
=
1
3. If
A
∩
B
= ∅
,
m
ʻ
(
A
∪
B
)
=
m
ʻ
(
A
)
+
m
ʻ
(
B
)
+
ʻ
m
ʻ
(
A
)
m
ʻ
(
B
)
.
Theorem 7.4.1
All mapping m
is, actually, a fuzzy measure.
ʻ
Proof
What lacks to be proven is that
A
ↂ
B
implies
m
ʻ
(
A
)
m
ʻ
(
B
)
. Since
A
C
A
C
A
ↂ
B
⃔
B
=
A
∪
(
∩
B
)
, with
A
∩
(
∩
B
)
= ∅
,itfollows
m
ʻ
(
B
)
=
A
C
A
C
A
C
A
C
m
ʻ
(
∩
B
)
+
ʻ
m
ʻ
(
A
)
m
ʻ
(
∩
B
)
=
m
ʻ
(
A
)
[
1
+
ʻ
m
ʻ
(
∩
B
)
]+
m
ʻ
(
∩
B
)
.
A
C
A
C
From
ʻ >
−
1, it follows 1
+
ʻ
m
ʻ
(
∩
B
)>
1
−
m
ʻ
(
∩
B
)
, and
m
ʻ
(
B
)>
A
C
A
C
m
ʻ
(
A
)
[
1
−
m
ʻ
(
∩
B
)
]=
m
ʻ
(
A
)
−
m
ʻ
(
A
)
m
ʻ
(
∩
B
)>
m
ʻ
(
A
)
.
1
−
m
ʻ
(
A
)
A
C
Theorem 7.4.2
m
ʻ
(
)
=
, for all A
∈ P
(
X
)
.
1
+
ʻ
m
ʻ
(
A
)
A
C
A
C
Proof
From
A
∩
= ∅
, follows
m
ʻ
(
X
)
=
1
=
m
ʻ
(
A
∪
)
=
m
ʻ
(
A
)
+
A
C
A
C
A
C
m
ʻ
(
)
+
ʻ
m
ʻ
(
A
)
m
ʻ
(
)
=
m
ʻ
(
)
[
1
+
ʻ
m
ʻ
(
A
)
]+
m
ʻ
(
A
).
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