数理解析研究所講究録1039 巻 1998 年 37-52

37

Representation ofComonotonically Additive Functional(共単調加法的汎関数の表現)

Yasuo NARUKAWA (成川康男), Toshiaki MUROFUSHI (室伏俊明),
Michio SUGENO (菅野道夫)
Dept. Comp. Intell. &Syst. Sci., Tokyo Inst. Tech.

1

Introduction

The Choquet integral with respect to a fuzzy measure proposed by Murofushi and$\mathrm{S}\mathrm{u}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{o}[6]$is a basic tool for subjectivegral is a functional on the class$B$$\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}[13]$and decision analysis [4]. This inte-of bounded measurable functions, which is monotone,

positive homogeneous and comonotonically additive (for short c.p.m.).
Conversely, concerning the problem of whether or not a c.p.m. functional on $B$ can be$I$

represented by a Choquet integral with respect to a fuzzy measure,that a c.p.m. functional

$I$

on

$B$

$\mathrm{S}\mathrm{c}\mathrm{h}\mathrm{m}\mathrm{e}\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e}\mathrm{r}[9]$provedcan be represented by a Choquet integral. Murofushi

et al. [8] proved that the comonotonically additive functional on$B$

can be represented bya Choquet integral with respect to a non-monotonic fuzzy measure.
Concerning the problem of whether or not a c.p.m. functional
by a Choquet integral when the domain of
$I$

$I$

is smaller than $B,$$I$

can be represented$\mathrm{G}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{o}[5]$proved it whenhas some continuity.
In this paper, we discuss fuzzy measures on a locally compact Hausdorff space $X$ and38

the Choquet integral with respect to a fuzzy measure. We discuss the functional definedon the class $K$ (or

on

$X$

$K^{+}$) of (resp. positive) continuous functions with compact support. We generally havebe a measurable set and$1_{A}\not\in K^{+}$$K^{+}\neq B^{+}\subset$$1_{A}$

on a locally compact Haus.

be the characteristic function of$\mathrm{d}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{f}\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{C}\mathrm{e}’\vee$$A$.

$\mathrm{I}.\mathrm{n}$fact, let$A$

, we have $1_{A}\in B^{+}$ but.

In section 2, basic properties of the fuzzy measure and the Choquet integral are shown,and we define the rank and sign dependent functional (for short r.s.d. functional), which$I$

is the difference of two Choquet integral. That is,$I(f)=(c) \int f^{+}d\mu^{+}-(c)\int f^{-}d\mu^{-}$

where $f^{+}=f\mathrm{O}$ and $f^{-}=-$ ( A ). This functional is used in cumulative prospect
$f$

$0$

theory [14].
In section 3, we define a regular fuzzy measure, and show its properties.It is shownthat if Choquet integrals of continuous functions with compact support with respect totwo regular fuzzy measures are equal to each other then the regular fuzzy measures areequal to each other. This means the uniqueness of regular fuzzy measure which represents

the$\mathrm{c}.\mathrm{m}$. functional.

In section 4, we show that the functional$I$

comonotonically additive and monotone (for short
$I$

on

$K$

$\mathrm{c}.\mathrm{m}.$is positive homogeneous if). We show that a$\mathrm{c}.\mathrm{m}$

$I$

is

. functionalcan be represented by a Choquet integral with respect to a regular fuzzy measure when

the domain of is the class$I$$K^{+}$of nonnegative continuous functions with compact support

on the locally compact Hausdorff space.
We show that the$\mathrm{c}.\mathrm{m}$. functional on$K$

is the r.s.d. functional in section 5. In section

6, we discuss the case of the universal set $X$ to be a compact Hausdorff space. It is shown

there that a$\mathrm{c}.\mathrm{m}$. functional can be represented by one Choquet integral.

39

2Preliminaries

In this section, we define fuzzy measure, the Choquet integral and the rank and signdependent functional, and show their basic properties. Throughout the paper we assume

that (X, ) be a measurable space.
$B$

A fuzzy measure

Definition 2.1 [11]$\mu:Barrow\overline{R^{+}}$

(1)

$\mu$

is an extended real valued set function,with the following properties.

$\mu(\emptyset)=0$(2) $\mu(A)\leq\mu(B)$ whenever $A\subset B$ ,where $A,$ $B\in B$ andWe say that

When

$\mu$

$\mu$

$\overline{R+}=[0, \infty]$

the set of extended nonnegative real numbers.

is finite if $\mu(X)<\infty$ .

is finite, we define the dual$\mu^{d}$

of

$\mu$

by

$\mu^{d}(A)=\mu(X)-\mu(A^{C})$for $A\in B$ .
We denote by$K_{0}$

the class of measurable functions and by$K_{0}^{+}$

the class of nonnegative

measurable functions.
Definition 2.2 $[1, 6]$ Let$\mu$

be a fuzzy measure on (X, ).

(1) The Choquet integral of$B$$f\in K_{0}^{+}$

with respect to a fuzzy measure,$(C) \int fd\mu=\int_{0}^{\infty}\mu_{f}(r)dr$

where $\mu_{f}(r)=\mu(\{x|f(X)\geq r\})$ .

$\mu$

is defined by

40(2) Suppose $\mu(X)<\infty$ . The Choquet integral of $f\in K_{0}$ with respect to a fuzzymeasure$\mu$

is defined by$(C) \int fd\mu=(C)\int f^{+}d\mu-(C)\int f^{-}d\mu^{d}$,where $f^{+}=f0$ and $f^{-}=-$ ( A ).
$0$

$f$

Definition 2.3 [3] Let $f,g$ be measurable nonnegative functions. We say thatare comonotonic if$f(x)<f(X’)\Rightarrow g(x)\leq g(X’)$

for$x,$ $x’\inX$. We denote $f\sim g$ , when

$f$

and

$g$

are comonotonic.

Theorem 2.4 $[2, 7]$ Let $f,$ $g\in K_{0}$ .
(1) If $f\leq g$ , then$(C) \int fd\mu\leq(c)\int gd\mu$(2) If$a$

is a nonnegative real number,$(C) \int afd\mu=a(C)\int fd\mu$(3) If $f,$$g\in K_{0}^{+}$.

are comonotonic, then$(C) \int(f+g)d\mu=(C)\int fd\mu+(C)\int gd\mu$ .

(4) If $\mu(X)<\infty$ and $f,$$g$

are comonotonic,$(C) \int(f+g)d\mu=(C)\int fd\mu+(C)\int gd\mu$ .

$f$

and

$g$

41

Definition 2.5 Let$I$

be a real-valued functional on$K\subset K_{0}.$$I$is said to be the rank

and sign dependent functional (for short the $r.s.d$ . functional) on $K$ , if there exist two

fuzzy measures$\mu^{+},$ $\mu^{-}$

such that for every $f\in K$$I(f)=(C) \int f^{+}d\mu^{+}-(c)\int f^{-}d\mu^{-}$

When$\mu^{+}=\mu^{-}$$\mu^{-}=(\mu^{+})^{d}$, the r.s.d. functional is the$\check{\mathrm{S}}\mathrm{i}_{\mathrm{P}}\mathrm{o}\check{\mathrm{S}}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}[10]$. If $\mu^{+}(X)<\infty$ and, the r.s.d. functional is the Choquet integral.

The regular fuzzy measure

3

In this section, we define the regular fuzzy measure and show some continuous propertiesof a regular fuzzy measure.
In the following we suppose that $X$ is a topological space and $K$ is the class of con-tinuous functions on $X$ with compact support. We denote supp $(f)$ the support of $f\in K$ ,that is,supp $(f)=cl\{X|f(_{X})\neq 0\}$ .
$||\cdot||$

on $K$ means the

We define$\sup$$K^{+},$ $K^{-},$ $K_{1}^{+}$

norm, and$cl(\cdot)$means th.e closure.

by$K^{+}=\{f|f\in K, f\geq 0\}$$K^{-}=\{f|f\in K, f\leq 0\}$

$K_{1}^{+}=\{f|f\in K, 0\leq f\leq 1\}$

Let

$B$

be the class of Borel subsets of $X,$

bf compact subsets of $X$ .

$\mathcal{O}$.

the class of open subsets of $X$ and$C$

the class42

Definition 3.1 Let
$\mu$

$\mu$

is a fuzzy measure on the measurable space(X, ).
$B$

is said to be outer regular if$\mu(B)=\inf\{\mu(O)|O\in \mathcal{O}, O\supset B\}$

for all $B\in B$ . An outer regular fuzzy measure$\mu$

is said to be regular if for all$\mu(O)=\sup\{\mu(c)|c\in C, C\subset O\}$

Definition 3.2 Let
$\mu$

$\mu$

be a fuzzy measure on the measurable space (X, ).
$B$

is said to be -continuous from below if$\mathit{0}$$O_{n}\uparrow O\Rightarrow\mu(O_{n})\uparrow\mu(O)$where $n=1,2,3,$$\mu$$\cdots$

and both$O_{n}$

and$O$

are open sets.

is said to be -continuous from above if$c$$C_{n}\downarrow C\Rightarrow\mu(C_{n})\downarrow\mu(C)$where $n=1,2,3,$$\cdots$

and both$C_{n}$and$C$

are compact sets.

The next two results follow from the definition.
Proposition 3.3 Let
(1)
(2)

$\mu$

be a regular fuzzy measure.

$\mu$

is -continuous from below.

$\mu$

is -continuous from above.

$\mathit{0}$$c$Corollary 3.4 Let$\mu$be a finite regular fuzzy measure.

(1)$\mu^{d}$is -continuous from below.

(2)$\mu^{d}$is -continuous from above.

$\mathit{0}$$c$.

$O\in \mathcal{O}$43

The next result follows from Proposition 3.3.
Theorem 3.5 Let $X$ be a locally compact Hausdorff space, and let

fuzzy measures. If $(C) \int fd\mu_{1}=(C)\int fd\mu_{2}$ for all $f\in K$ , then
Even if$\mu^{d}$$\mu$

is regular, it is possible that$\mu^{d}$$\mu_{1}$and$\mu_{1}=\mu_{2}$$\mu_{2}$be regular.

is not regular. But Corollary 3.4 shows thatis -continuous from above and -continuous from below. Therefore we can obtain the$c$$\mathit{0}$next corollary.
Corollary 3.6 Let$\mu$be a finite regular fuzzy measure.

If $(C) \int fd\mu=(C)\int fd\mu^{d}$ for all $f\in K$ then $\mu(C)=\mu^{d}(C)$ where

4

Representation of the$\mathrm{c}.\mathrm{m}$$C$.functional on

We assume that $X$ is a locally compact Hausdorff space and$B$is compact.

$K^{+}$

the class of its Borel

subset. Let $K$ be the set of continuous functions with compact support, and$K^{+}$

the setof continuous nonnegative functions with compact support.
Definition 4.1 Let$I$

be a real valued functional on $K$ . We say that

additive iff $f\sim g\Rightarrow I(f+g)=I(f)+I(g)$ for $f,$ $g\in K^{+},$$I(af)=aI(f)$ for all positive real
$I(g)$

for $f,$ $g\in K^{+}$ . If the functional

that

$I$

Let

, and

$\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}a\backslash >0$

$I$

$I$

$I$

be a

$\mathrm{c}.\mathrm{m}$is comonotonicallyis positively homogeneous iff

is monotone iff $f\leq g\Rightarrow I(f)\leq$

is comonotonically additive and monotone, we say

is a $c.m$ . functional.
$I$

$I$

. functional on $K$ . It follows from $f\sim f$ that
$I(nf)=nI(f)$and$nI( \frac{1}{n}f)=I(f)$44

for $f\in K$ and a positive integer . Then we have $I(qf)=qI(f)$ if is a positive rational$n$number. Let

$P$

$q$

be a positive real number. For every positive integer , there exists a$n$rational number$r$such that $r<P<r+ \frac{1}{n}$ . Since$I$

is monotone, we have the next

proposition.

Proposition 4.2 A $c.m$ . functional on $K$ is positive homogeneous.

In this section, we shall demonstrate that iffunctional

$I$

on

$K^{+}$$I$is a$\mathrm{c}.\mathrm{m}$. functional then a real-valued

is represented as a Choquet integral with respect to a regular fuzzymeasure.
Lemma 4.3 Let I be a $c.m$ . functional on$K^{+}$. We put$\mu(O)=\sup\{I(f)|f\in K_{1}^{+}, \mathit{8}upp(f)\subset O\}$,and$\mu(B)=\inf\{\mu^{+}(O)|O\in \mathcal{O}, O\supset B\}$

Then

$\mu$

.

is an outer regular fuzzy measure.

We shall say that this fuzzy measure$\mu$

is the fuzzy measure induced by ational .
$I$

Proposition 4.4 Let$\mu$is the fuzzy measure induced by a $c.m$ . functional .
$I$(1) If $f\in K^{+},A\subset\{x|f(X)\geq 1\}$ and $A\in B$ , then $\mu(A)\leq I(f)$ .
(2) If$C$

is a compact set in(3) If$O$

is an open set in $B,$ $\mu(O)=\sup\{\mu(C)|c\in C, C\subset O\}$ .

$B$, then $\mu(C)<\infty$ .

$\mathrm{c}.\mathrm{m}$. func-

45

It follows from Propostion 4.4 that the fuzzy measure induced by a$\mathrm{c}.\mathrm{m}$. functional$I$

is regular.
Theorem 4.5 For a $c.m$ . functional I onon$K^{+}$, there exists a regular fuzzy measure$\mu$such that for all $f\in K^{+}$$B$$I(f)=(C) \int fd\mu$

proof) Let $f\in K^{+},$ $O_{n,k}=\{x|f(x)>(k-1)/n\}$ and $C_{n,k}=\{x|f(x)\geq k/n\}$ where

1 $\leq k\leq n$ . Then for all

$n$

and

$O_{n,k+1}\subset C_{n,k}\subset O_{n,k}\subset supp(f)$

all

$n,$

supp

$k$

$k,$

$C_{n,k}$. Sinceis a compact set,$X$

$O_{n,k}$is an open set, andis a locally compact Hausdorff space, forthere exists $f_{n,k}\in K^{+}$ such that $0\leq f_{n,k}\leq 1,$ $f_{n,k}(X)=1$ when $x\in C_{n,k}$ and$(f_{n,k})\subset O_{n,k}$.

These functions$f_{n,k}$have the following properties.

(1) For all positive integer $n,$

$k$

and

$j$

such that $1\leq k\leq n$ and $1\leq j\leq n,$$f_{n,k}$and$f_{n,j}$are comonotonic.
(2) For all positive integer$f_{n,1}+f_{n}:^{2}+\cdots$$n$$+f_{n,k}$

and

$k$

such that $1\leq k\leq n$ ,and $f_{n,k+1}+f_{n,k+2}+\cdots$$+f_{n,n}$

are comonotonic.

Next define $f_{n}\in K^{+}$ by$f_{n}-- \sum_{=k1}^{n}\frac{1}{n}f_{n},k$for $n=1,2,$$\cdots$. If $k/n\leq f(x)<(k+1)/n$ , then $k/n\leq f_{n}(x)\leq(k+1)/n$ since$x\in C_{n,k}\subset C_{n,k-1}\subset\cdots\subset C_{n,1}$

obtain $||f-f_{n}||\leq 1/n$ , where

and $x\not\in C_{n,k+1}\supset O_{n,k+2}\supset\cdots\supset O_{n,n}$ . Therefore we$||\cdot||$is the $\sup$ norm.

Then there exists $F\in K^{+}$ which satisfies the following conditions.
(1) $0\leq F\leq 1$

46(2) $f_{n}\sim F$ for all$n$(3) $x\in supp(f)\Rightarrow F(x)=1$(4) $0 \leq|f-f_{n}|\leq\frac{1}{n}F$

Therefore we have$\lim_{narrow\infty}I(f_{n})=I(f)$.

$1$

Since the Choquet integral with respect to every fuzzy measure is a c.p.m. functional,

we can obtain the following corollary.
Corollary 4.6 For every fuzzy measure , there exists an outer regular fuzzy measure$\mu$$\mu_{r}$such that for every $f\in K^{+}$$(C) \int fd\mu=(c)\int fd\mu_{r}$Example 1 Let $X=[0,1],$$B$

We define a fuzzy measure.

the family of Borel subsets of $X$ and

$\mu$

on

$B$

$J=[0, \frac{1}{2}]$

.

by
1

if

$A\not\subset J$

$0$

if

$A\subset J$$\mu(A)=\{$

Then this fuzzy measure$J\subset O$$\mu$

is not outer regular. In fact, let$O$be an open set such that

. Since is not open, $J\neq O$ . Therefore we obtain $\mu(O)=1,$ $\inf\{\mu(O)|0\supset J\}=1$ ,$J$

and $\mu(J)=0$ .

We define an outer regular fuzzy measure by
1

if$A\not\subset J’$

$0$

if

$A\subset J’$$\mu_{r}(A)=\{$

where $A\in B$ and$J’=(0, \frac{1}{2})$. Then for all $f\in K^{+},$ $(C) \int fd\mu=(C)\int fd\mu_{r}$ .

47

5Representation of the$\mathrm{c}.\mathrm{m}$.functional on $K$

In this section, we discuss the functional defined on the class $K$ of continuous functions

with compact suport. We define the regular fuzzy measure induced by aon $K$ , and show that a$\mathrm{c}.\mathrm{m}$$\mathrm{c}.\mathrm{m}$. functional. functional $K$ can be represented by regular fuzzy measures.

We obtain the next lemma from Proposition4.4.
Lemma 5.1 Let I be a $c.m$ . functional on $K$ .
(1) We put$\mu^{+}(O)=\sup\{I(f)|f\in K_{1}+,psup(f)\subset O\}$,and$\mu^{+}(B)=\inf\{\mu^{+}(O)|\mathit{0}\in \mathcal{O}, \mathit{0}\supset B\}$for$O\in \mathcal{O}$and $B\in B$ . Then$\mu^{+}$

is a regular fuzzy measure.

(2) We Put$\mu^{-}(o)=\sup\{-I(-f)|f\in K_{1}+,u\mathit{8}pp(f)\subset O\}$,and$\mu^{-}(B)=\inf\{\mu^{-}(O)|O\in \mathcal{O}, O\supset B\}$for$O\in \mathcal{O}$and $B\in B$ . Then

Definition 5.2 Let

$I$

be a

$\mathrm{c}.\mathrm{m}$$\mu^{-}$

is a regular fuzzy measure.

. functional on $K$ . We say thatis the regular fuzzy measure induced by$I^{+}$, and$\mu^{-}$$\mu^{+_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}}}\mathrm{n}\mathrm{e}\mathrm{d}$in Lemma 5.1

the regular fuzzy measure induced by$I^{-}$

It follows from Proposition 3.3 that the induced regular fuzzy measures

are o–continuous from below and -continuous from above.
$c$

We obtain the next lemma from Theorem 4.5.

$\mu^{+}$

and$\mu^{-}$

48

Lemma 5.3 Let I be a $c.m$ . functional on $K$.
(1) If$\mu^{+}$

is the regular fuzzy measure induced by$I^{+}$, then we have$I(f)=(C) \int fd\mu^{+}$

for all(2) If$\mu^{-}$$f\in K^{+}$.

is the regular fuzzy measure induced by$I^{-}$, then we have$I(f)=-(C) \int(-f)d\mu-$for all$f\in K^{-}$

Let $f\in K$ and be a$I$$\mathrm{c}.\mathrm{m}$. functional on$K$. Since$f\vee \mathrm{O}\sim f$

AO and$f=(f\vee \mathrm{o})+(f\wedge \mathrm{o})$,we have$I(f)=I(f\mathrm{O})+I$ ( $f$ A

Since

$f$

VO

$\in K^{+}$

and

$f$

$0$

).

A $0\in K^{-}$ , the next theorem follows from Lemma 5.1 and Lemma

5.3.
This is the main result in this section.
Theorem 5.4 A $c.m$ . functional on $K$ is a $r.s.d$ . functional. That is, let

the regular fuzzy measures induced by$I^{+}$

and$I^{-}$$\mu^{+}$

and$\mu^{-}$

berespectively. We have$I(f)=(C) \int(f\vee 0)d\mu^{+}-(C)\int-$ ( $f$ A$0$)$d\mu^{-}$

for every $f\in K$ .
The next corollary gives the nesessary and sufficient condition that a$I$$\mathrm{c}.\mathrm{m}$. functional

is the \v{S}ipo\v{s} integral.

Corollary 5.5 Let I be a $c.m$ . functional on K. If $I(f)=-I(-f)$ for every $f\in K$ ,then I is the \v{S}ipo\v{s} integral.

49

It seems that$\mu^{+}=\mu^{-}$

and$\overline{\mu^{+}}=\mu^{-}$

if the functional

not always true. The next example shows that$\mu^{+}\neq\mu^{-}$

$I$

is a

and

$\mathrm{c}.\mathrm{m}$$(\mu^{+})^{d}\neq\mu^{-}$

Example 2 Let $X=R$ and $J=[0,1]$ . Define a fuzzy measure

1 if

or

$J\subset A$$A$. functional. But it is$\mu$: $Barrow\{0,1\}$ by

is not bounded$\mu(A)=\{$
$0$

if

$A$

is

$\mathrm{b}\dot{\mathrm{o}}$unded and$J\cap A^{c}\neq\emptyset$

for $A\in B$ . And define a functional on $K$ by$I(f)=(c) \int fd\mu$

Then

is a

$I$

induced by$\mathrm{c}.\mathrm{m}$$I^{+}$.

. functional. Therefore there exist two regular fuzzy measureand$\mu^{+}$

and$\mu^{-}$. Indeed for $A\in B$$I^{-}\mathrm{r}\mathrm{e}\mathrm{s}_{\mathrm{P}^{\mathrm{e}\mathrm{c}}}\mathrm{t}\mathrm{i}_{\mathrm{V}\mathrm{e}1}\mathrm{y}$1 if$J\subset A$$\mu^{+}(A)=\{$

if

$0$

$J\cap A^{c}\neq\emptyset$

and$\mu^{-}(A)=0$

whenever$A\in B$.

Then we have

1 if

$J\cap A\neq\emptyset$$(\mu^{+})^{d}(A)=\{$
$0$

and

if

$A\subset J^{c}$Thus$\mu^{+}\neq\mu^{-}$

6

The case of compact Hausdorff space$(\mu^{+})^{d}\neq\mu^{-}$

If $X$ is a locally compact Hausdorff space, $\mu^{+}(X)=\mu^{-}(X)$ is not always true and $I(f)$is not always equal to the Choquet integral of$f$with respect to, we suppose that $X$ is a compact Hausdorff space.

$.\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}_{0}\mathrm{n}$$\mu^{+}$. Throughout this50

If $X$ is compact, then the class $K$ of continuous functions on $X$ with compact support

is the class of continuous functions on $X$ . Since we have $1_{X}\in K$ , we have $\mu^{+}(X)=I(1_{X})$and $\mu^{-}(X)=-I(-1_{x)}$ . The next result follows immediately from this fact.
Proposition 6.1 Let I be a $c.m$ .

measure induced by
Let$f\in K$

$I$

be a

$\mathrm{c}.\mathrm{m}$$I^{+}$

and$I^{-}$

functional on

$K$

and

$\mu^{+}$

and$\mu^{-}$

the regular fuzzy

respectively. Then we have $\mu^{+}(X)=\mu^{-}(X)$ .

. functional on $K,$$\mu^{+}$

the regular fuzzy measure induced by$I^{+}$

and. There exists $a>0$ such that $||f||<a$ . It follows from Theorem 5.4 that
$I(f+a1_{X})=(C) \int(f+a1_{X})d\mu^{+}$ .

Since $f\sim a1_{X}$ , we have the next theorem.
Theorem 6.2 Let I be a $c.m$ . functional on $K$ andby$I^{+}$$\mu^{+}$

the regular fuzzy measure induced. Then I can be represented by the Choquet integral with respect to$\mu^{+}$. That is,$I(f)=(c) \int fd\mu^{+}$

for $f\in K$ .
The proof of the next corollary is much the same.
Corollary 6.3 Let I be a $c.m$ . functional on $K$ , andby$I^{-}$$\mu^{-}$

the regular fuzzy measure induced

Then we have$I(f)=-(C) \int(-f)d\mu-$

for $f\in K$ .
The next result follows immediately from Theorem 5.4, Theorem 6.2 and Corollary

3.6.

51

Corollary 6.4 Let I be a $c.m$ . functional on $K$ andinduced by$I^{+}$$\mu^{+}$

and$\mu^{-}$

the regular fuzzy measure

and $I^{-}reSpectively$ .

(1) $(C) \int fd\mu^{-}=(C)\int fd(\mu^{+})^{d}$ for all $f\in K^{+}$ .
(2)$\mu^{-}=(\mu^{+})^{d}$and$\mu^{+}=(\mu^{-})^{d}$.

Corollary 6.4 (2) means that both$\mu^{+}$

and$(\mu^{+})^{d}$($\mu^{-}$

and$(\mu^{-})^{d}$) are regular when $X$

is compact.

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