数理解析研究所講究録 1253 巻 2002 年 73-86

73

Representation of Choquet Integral
桐朋学園

成川康男 (Yasuo NARUKAWA)

Toho Gakuen ,東工大・総理工室伏俊明 (Toshiaki MUROFUSHI),

Dept. Comp. Intell. &Syst. Sci., Tokyo Inst. Tech.

1

Introduction

Non-additive set functions on measurable space is used in economics, decision theory
and artificial intelligence, called by various name, such as cooperative game , capacity

or fuzzy measure. In this paper, according to Denneberg [4] we call them non-additive

measures. The Choquet integral with respect to anon-additive measure is abasic tool
for multicriteria decision making, image processing and recognition $[8, 9]$ . We consider
the space$\mathcal{F}\mathcal{M}^{+}$of non-additive measures with topology introduced by Choquet integral.

The subspace$\mathcal{F}\mathcal{M}_{1}^{+}$of non-additive measures

universal set is compact. The space$\mathcal{F}\mathcal{M}^{+}$$\mu$satisfying $\mu(X)=1$ where $X$ is theof non-additive measures is alocally convex

space. Applying the facts mentioned above, we obtain the additive representation of
Choquet integral, that is, the Choquet integral with respect to anon-additive measure isrepresented by the classical integral with respect to the classical measure.
The similar theorems are shown in various contexts. In $[11, 12]$ , Murofushi and Sugeno
show the additive representation theorem and propose an interpretation that non-additive

measures express with their non additivity interaction among subset. Denneberg [5] show$\mathrm{s}$74
the additive representation theorem, that is ageneralization in various fields of papers,such as Gilboa and Schmeidler [6, 7] and Marinacci [10].

We compare these representation theorems, and show the equivalent points and thedifference among them. We show that the domain of the representing classical measure
and the representing integrand of classical integral are equivalent, but the classical mea-sures which represent the non-additive measures are different.

2Non-additive measure and Choquet integral
In this subsection, we present basic definitions and theorems about non-additive measuresand the Choquet integral.

Definition 2.1. Let

be ameasurable space. Anon-additive measure$(X, \mathcal{X})$real valued set function,

$\mu$

:

$\mathcal{X}arrow R^{+}$(ii) $\mu(A)\leq\mu(B)$ whenever A$\subset B$, A, B

with the following properties: (i)$\in \mathcal{X}$$\in \mathcal{X}$is an$\mu(\emptyset)=0$

and, where $R^{+}=[0,\infty)$ is the set of extended

nonnegative real numbers. We define the conjugate
A

$\mu$

$\mu^{c}$ofby $\mu^{c}(A)=\mu(X)-\mu(A^{c})$ for

$\mu$

.

The class of bounded measurable functions is denoted by

non negative measurable functions by

Definition 2.2. $[1, 11]$ Let$\mu$$\mathcal{L}^{\infty+}$$\mathcal{L}^{\infty}$.

be anon-additive measure on(1) The Choquet integral of f$\in \mathcal{L}^{\infty+}$with respect to$\mu$$(X,\mathcal{X})$.

is defined by(C) $\int fd\mu=\int_{0}^{\infty}\mu_{f}(r)dr$ ,where $\mu_{f}(r)=\mu(\{x|f(x)\geq r\})$ .

and the class of bounded75(2) Suppose $\mu(X)<\infty$ . The Choquet integral ofwith respect to$f\in \mathcal{L}^{\infty}$$\mu$

is definedby(C) $\int fd\mu=(C)\int f^{+}d\mu-(C)\int f^{-}d\mu^{\mathrm{c}}$ ,where $f^{+}=f\vee 0$ and $f^{-}=-(f\Lambda 0)$ .

Definition 2.3. Anon monotonic non-additive measure$A_{1}$,$\mu$

is -monotone $(k\geq 2)$ if for$k$$\cdots A_{k}\in \mathcal{X}$$\mu(\cup^{k}A_{i})+\sum_{Ii=1\mathrm{C}1,\cdots,k,I\neq\emptyset}(-1)^{|I|}\mu(\bigcap_{I}A_{i})\geq 0i\in$

We say thatis totally monotone if it is monotone and -monotone for all $k\geq 2$ .
$k$

$\mu$

If $\mu(X)=1$ andLet$\mathcal{F}\mathcal{M}^{+}$

$\mu$

$f$

is totally monotone,$\mu$

is abelief function.

be the class of (monotone) non-additive measures. We define$\{\mu-\nu|\mu, \nu \in \mathcal{F}\mathcal{M}^{+}\}$

Let

.

. and

$\mathcal{F}\mathcal{M}^{1}:=\{\mu\in \mathcal{F}\mathcal{M}^{+}|\mu(X)=1\}$.

be anonnegative measurable function. We define the map$C_{f}( \mu):=(C)\int fd\mu$. We define

for$C_{A}=C_{1_{A}}$nonnegative measurable functions by$B^{+}$$X\in \mathcal{X}$$\mathcal{F}\mathcal{M}:=$$C_{f}$:$\mathcal{F}\mathcal{M}arrow R$

by. We denote the set of bounded. It is obvious that

$C_{f}$is alinear map on$\mathcal{F}\mathcal{M}$for all $f\in B^{+}$ .

Definition 2.4. We shall say that the coarsest topology for which every
for$A\in \mathcal{X}$is$\mathcal{X}$-topology forcontinuous for $f\in B^{+}$ isDefinition 2.5. Let$E$$B^{+}$$\mathcal{F}\mathcal{M}$-topology for$\mathrm{c}(\mathrm{A})$$c(A)=\cap$is continuous, and that the coarsest topology for which every$\mathcal{F}\mathcal{M}$.

be avector space and $A\subset E$ .

We define the convex hull

$C_{A}$

by{, Yis aconvex set}$\mathrm{Y}|A\subset \mathrm{Y}$$C_{f}$

is76
We say that x $\in X$ is an extreme point of X if x$=\lambda x_{1}+(1-\lambda)x_{2};x_{1}$implies $x_{1}=x_{2}=x$ . We denote the set of extreme points of A by

Definition 2.6. We say that$\mu(A)=1$ for all$A\in \mathcal{X}$$\mu\in \mathcal{F}\mathcal{M}^{1}$, $x_{2}\in X,0\leq\lambda$$\mathcal{E}(A)$$\leq 1$.

is 0–1 non-additive measure if $\mu(A)=0$ or. We denote the set of 0-1 non-additive measures by$\mathcal{F}\mathcal{M}_{0}^{1}$. That1s,:$\mathcal{F}\mathcal{M}_{0}^{1}=\{\mu|\mu\in \mathcal{F}\mathcal{M}^{+},\mu$ $\mathcal{X}arrow\{0,1\}\}$.

3Integral representations
In this section, we show three integral representation theorem.

3.1

Representation by Choquet theorem

In this subsection, we show the representation theorem of Choquet integral by topologicalapproach. The details of the proofs are in [13].

Let

$E$

be aHausdorff localy convex space andof continuous convex function on$\mathrm{Y}$is denoted by$\mathrm{Y}\subset E$$S(\mathrm{Y})$Let $K(E, R)$ be the class of continuous functions

be compact and convex. The set. Define$f$$A(\mathrm{Y}):=S(\mathrm{Y})\cap(-S(\mathrm{Y}))$.

: $Earrow R$ with compact support.

On $K(E, R)$ we put the order defined by $f\geq 0$ if and only if $f(x)\geq 0$ for all $x\in E$ .
Aradon measure on$E$

is alinear map

compact there exists anumber$\mu(f)\leq M_{\mathrm{Y}}||f||$$M_{\mathrm{Y}}$$\mu$: $K(E, R)arrow R$ such that for any

such that $f\in K(E, R)$ and supp(f), where supp(f) is asupport of$f$and the norm

The collection of Radon measures on $E$ is denoted bymeasures with the order defined byby$\mathcal{R}^{+}(E)$. We define the order$\mu\geq 0$$\prec \mathrm{i}\mathrm{n}$$\mathcal{R}(E)$$||\cdot$$||$$\subset \mathrm{Y}$$\mathrm{Y}\subset E$

impliesis the $\sup$ norm.

and the set of positive Radon

if and only if $\mu(f)\geq 0$ for all $f\geq 0$ , $f\in K(E, R)$$\mathcal{R}^{+}(\mathrm{Y})$by$\mu\prec\nu$

if and only if $\mu(f)\leq\nu(f)$ for

77
all

$f\in S(\mathrm{Y})$. There exists amaximal elementthat the maximal element$||\mu||$$f$$m\in \mathcal{R}^{+}(\mathrm{Y})$$m\in \mathcal{R}^{+}(\mathrm{Y})$with respect

is amaximal measure. For$:= \sup\{|\mu(f)||f\in K(E, R), ||f||\leq 1\}$

. We define$\mathcal{R}^{1}$$\mathrm{t}\mathrm{o}\prec$$\mu\in \mathcal{R}(E)$. We say, we define$:=\{\mu\in \mathcal{R}^{+}(E)|||\mu||=1\}$. Let

be abounded real-valued function. Define $\hat{f}(\mathrm{Y}):=\inf\{g(x)|g\in(-S(\mathrm{Y})),g\geq f\}$ . We

say that$\mathrm{Y}_{f}$is the bordering set of

The space$\mathcal{F}\mathcal{M}$

with

if

$\mathrm{Y}_{f}=\{x\in \mathrm{Y}|f(x)=\hat{f}(x)\}$-topology is aHausdorff locally convex space, and$B^{+}$

is compact convex. Define$f$$h_{f}$:$\mathcal{F}\mathcal{M}^{1}arrow R$

by $h_{f}(\mu)=C_{f}(\mu)$ for $f\in B^{+}$ . Then$\mathcal{F}\mathcal{M}^{1}$$h_{f}$

islinear and continuous. Applying Choquet theorem $[3, 2]$ there exists amaximal radonmeasure$m\in \mathcal{R}(\mathcal{F}\mathcal{M}^{1})$$\mathcal{F}\mathcal{M}^{1}|\mu(A)=0$

such that $h_{f}=m(hf)$ andor $\mu(A)=1\}$ ,$A\in \mathcal{X}$$m(\mathcal{F}\mathcal{M}^{1}\backslash G_{A})=0$

for $G_{A}:=\{\mu\in$.

Applying Riesz’s Representation theorem we have the next theorem.
Theorem 3.1. For every$\mu\in \mathcal{F}\mathcal{M}^{1}$, there exists a maximal Radon measure$m\in \mathcal{R}^{1}$such that(C) $\int fd\mu=\int h_{f}dm$ ,for allif$f\in B^{+}$$\mathcal{F}\mathcal{M}^{1}$and$m(\mathcal{F}\mathcal{M}^{1}\backslash G_{A})=0$

for every$A\in \mathcal{X}$. Especially$m(\mathcal{F}\mathcal{M}^{1}\backslash \mathcal{F}\mathcal{M}_{0}^{1})=0$

is metrizable.

We say that $(hf, m)$ is Choquet representation for $(f, \mu)$ .

As to the metrizability of

Proposition 3.2. If$B^{+}$$\mathcal{F}\mathcal{M}^{1}$

we have the next proposition.

is separable, the$\mathcal{F}\mathcal{M}^{1}$is separable and metrizable.

Next we consider the uniqueness of Choquet representation. First we define the ChO-

quet simplex. Let

and$\hat{\mathrm{Y}}=\tilde{\mathrm{Y}}-\tilde{\mathrm{Y}}$$\mathrm{Y}\subset E$

be convex and compact. DenoteWe say that$\mathrm{Y}$$\tilde{\mathrm{Y}}=\{(\lambda x, \lambda)/x\in \mathrm{Y}, \lambda>0\}$is Choquet simplex if there exists $\sup(x_{1}, x_{2})$ for78$x_{1}$,$x_{2}\in\hat{\mathrm{Y}}$, where the orderdefined by$\prec \mathrm{i}\mathrm{s}$$x_{1}\prec x_{2}$if and only if$x_{2}-x_{1}\in \mathrm{Y}$. It fol-lows Choquet theorem [3] that the Choquet representation is unique if and only if
is Choquet simplex. But$\mathcal{F}\mathcal{M}^{1}$$\mathcal{F}\mathcal{M}^{1}$

is not always Choquet simplex. Therefore the Choquet

representation is not always unique.

3.2

Interpreter representation

In this subsection, according to Murofushi and Sugeno we present the interpreter repre

sentation theorem. All proofs are shown in [11, 12].

Definition 3.3. Let$(X, \mathcal{X})$(1) Amapping $H$ :Let$\mathcal{X}arrow \mathcal{Y}$$(\mathrm{Y}, \mathcal{Y})$be measurable spaces.

is called an interpreter from

$H(\emptyset)=\emptyset$, (b) $H(A)\subset H(B)$ whenever $A\subset B$ .

Atriplet$(\mathrm{Y},\mathcal{Y}, H)$$(X, \mathcal{X},\mu)$is called aframe of$(\mathrm{Y},\mathcal{Y})$$(X,\mathcal{X})$$\mathcal{X}$

$\mu$

to

$\mathcal{Y}$if $H$ satisfies (a)if $H$ is an interpreter frombe anon-additive measure space. Aquadruplet

interpreter representation ofonand

if $H$ is an interpreter from$\mathcal{X}$$\mathcal{X}$$\mathcal{Y}$.

is called an$(\mathrm{Y}, \mathcal{Y}, m, H)$to)), $m$ is aclassical measure

and $\mu=m\mathrm{o}H$ .

Asemifilter 0in ameasurable space
properties ;(1)$\emptyset\not\in\theta$, (2) if

$A\in\theta$

andDenote the set of all semifilters in$(X, \mathcal{X})$is anon empty subclass of$A\subset B\in \mathcal{X}$$(X, \mathcal{X})$by$S_{X}$

then$B\in\theta$$\mathcal{X}$with the.

, and define amapping$H_{X}$:$\mathcal{X}arrow 2^{S_{X}}$by $H_{X}(A):=\{\theta\in S_{X}|A\in\theta\}$ .
$Sx$

to

denotes the a-algebra generated by $\{H\chi(A)|A\in \mathcal{X}\}$ .

The triplet $(Sx,Sx, H_{X})$ is called the universal frame of$(X, \mathcal{X})$for representation

79
Theorem 3.4. For every non-additive measure$\mu$on (X,$\mathcal{X})$there exists a classical mea-

sure $m$ on $Sx$ such that $(S_{X}, S_{X}, m, H_{X})$ is a representation of .
$\mu$

Definition 3.5. Let
sure space$(X, \mathcal{X},\mu)$$(\mathrm{Y}, \mathcal{Y}, m, H)$

be an interpreter representation of anon-additive mea-.

For anon negative measurable function$i_{f}(y):= \sup\{r|y\in H(\{f\geq r\})\}$. We call$i_{f}$$f$on $X$ we define afunction$i_{f}$

on$\mathrm{Y}$byan interpreter for ameasurable function$f$induced by $H$ .
Theorem 3.6. Let

sure space$(X, \mathcal{X},\mu)$$(\mathrm{Y}, \mathcal{Y}, m, H)$

and

$i$

be an interpreter representation of a non-additive mea-

be an interpreter induced by H. We have(C) $\int fd\mu=\int i_{f}dm$ .

for$f\in \mathcal{L}^{\infty+}$.

We have the next theorem from Theorem 3.4 and Theorem 3.6.
Theorem 3.7. (Interpreter representation theorem) Let (X,$\mathcal{X}, \mu)$be a non-additive mea-

sure space. There exists a classical measure $m$ on $Sx$ such that(C) $\int fd\mu=\int i_{f}dm$

for$f\in \mathcal{L}^{\infty+}$

3.3.

Representation with M\"obius transformIn this subsection, we present the representation with Mobius transform by Denneberg.

The essence of the proofs are shown in [5].
$\mathcal{F}\mathcal{M}_{0}^{1}$denotes the set of 0–1 non-additive measures. We define the tilde operator

which assigns to ameasurable function$f$: $Xarrow[0, \infty]$ the function$\tilde{f}(\eta):=(C)[$ $fd\eta$,$\eta\in \mathcal{F}\mathcal{M}_{0}^{1}$

.

80
If

$A\in \mathcal{X},\tilde{A}$is defined by$\overline{\mathcal{T}}:=\{\tilde{A}|A\in \mathcal{T}\}$, for aclass$\tilde{A}:=\{\eta\in \mathcal{F}\mathcal{M}_{0}^{1}|\eta(A)=1\}$. We use the notation.

$\mathcal{T}\subset 2^{X}$

Definition 3.8. Anon-additive measure$u_{A}\in \mathcal{F}\mathcal{M}_{0}^{1}$

1

defined by$A\subset B$$u_{A}(B)=\{$

0$0.\mathrm{w}$.

is called aunanimity game for coalition A.
In some literature, it is called a0-1 necessity measure.

We write the set of all unanimity games on$\mathcal{X}$by$\mathcal{F}\mathcal{M}_{0u}^{1}$.

$\mathcal{F}\mathcal{M}_{0s}^{1}$denotes the set of

all supermodular 0-1 non-additive measures, that is,$\eta\in \mathcal{F}\mathcal{M}_{0s}^{1}\Leftrightarrow\eta(A\cup B)+\eta(A\cap B)\geq\eta(A)+\eta(B)$

for

$A$

,

$B\in \mathcal{X}$It is obvious that$\mathcal{F}\mathcal{M}_{0u}^{1}\subset \mathcal{F}\mathcal{M}_{0s}^{1}\subset \mathcal{F}\mathcal{M}_{0}^{1}$.

Let $f:arrow R$ be ameasurable function. We denote by$\{x\in X|f(x)\geq\alpha\}$$2^{\mathcal{F}\lambda 4_{0}^{1}},2^{\mathcal{F}\mathcal{M}_{0u}^{1}}$We use$D$,$D_{u}$

orand$\mathcal{F}\mathcal{M}_{0}^{1}$$D_{s}$.

$\subset 2^{F\mathcal{M}_{0}^{1}}$$2^{\mathcal{F}\mathcal{M}_{0s}^{1}}$,$D_{u}\subset 2^{\mathcal{F}F\mathrm{I}_{0u}^{1}}$

(1)

and

$D_{s}\subset 2^{F\mathcal{M}_{0\epsilon}^{1}}$the algebra generated by

as avariable for one of the sets$\mathcal{F}\mathcal{M}_{0}^{1},\mathcal{F}\mathcal{M}_{0u}^{1}$or$\mathcal{F}\mathcal{M}_{0s}^{1}$, and D. as.
$\kappa$for$\mathcal{X}$

is afunction$\kappa$:$\mathcal{F}\mathcal{M}_{0}^{1}$.

$\cross \mathcal{X}arrow[0, b]$such that$\kappa_{\eta}$$\tilde{X}$respectively.

Definition 3.9. Akernel function$\kappa_{\eta}(A)$the class of upper level sets.

We denote by $D$in$\mathcal{M}_{f}$is anon-additive measure on(2) For fixed A$\in \mathcal{X}$$\mathcal{X}$, the real function,$\eta\in \mathcal{F}\mathcal{M}_{0}^{1}.$$\kappa.(A)$on.

$\mathcal{F}\mathcal{M}_{0}^{1}$.

is Immeasurable;$(\eta, A)\mapsto*$81
Next we define x-extension.

Definition 3.10. Let$f\in \mathcal{L}^{\infty}$

and$\kappa$

be akernel function. -extension$\kappa$$f^{\kappa}( \eta):=(C)\int f(x)d\kappa_{\eta}(x)$

Let

$\nu$

be anon-additive measure on D%. We define the$\mu(A):=\int\kappa_{\eta}(A)d\nu(\eta)$

If$\kappa_{\eta}(A)=\eta(A)$functionof$\nu$$\kappa_{\eta}=\eta$

on

for

$(\eta, A)\in \mathcal{F}\mathcal{M}_{0}^{1}$$\mathcal{F}\mathcal{M}_{0u}^{1}$.

$\cross D$, then$\kappa$,$\kappa$of$f^{\kappa}$

defined by.
transform$A\in \mathcal{X}$

$\mu$

on

$\mathcal{X}$$\mathcal{X}$

of

by

$\nu$

.

is akernel function foris called the zeta function for$f$$\mathcal{X}$. The kernel. The corresponding transformis called the zeta transform of .
$\nu$

The next proposition is Example 4.1 in Denneberg [5].

Proposition 3.11. Let$\kappa$

be a zeta function. Then we have$f^{\kappa}=\tilde{f}$for$f\in \mathcal{L}^{\infty}$.

The next theorems are the main theorems in this subsection.
Theorem 3.12. Let$\kappa_{i}$be kemel functions for [andtions on D., $i=1,2$ respectively . Let jiij the$\kappa:=\kappa_{1}-\kappa_{2},$$\nu:=\nu_{1}-\nu_{2}$$k_{i}$$\nu_{i}$monotone and additive set func-

transform of$\nu j$and $\mu:=(\mu_{11}-\mu_{12})-(\mu_{21}-\mu_{22})$ . If

for $i,j=1,2$ . Define$f\in \mathcal{L}^{\infty}$, then$f^{\kappa}--f^{\kappa_{1}}-f^{\kappa_{2}}\in \mathcal{L}^{\infty}$

and(C) $\int fd\mu=\int f^{\kappa}d\nu$ .
$\nu$

,

$\mu$

and

$\kappa_{\eta}$

are non monotonic non-additive measures of bounded variation for

Corollary 3.13. Let

Let

$\mu$

$k$

be a zeta function forbe the zeta transform of . If f
$\nu$

(C)

$\in \mathcal{L}^{\infty}$$\mathcal{X}$

and

$\nu$

, then$\int fd\mu=\int\overline{f}d\nu$.

$\mathcal{X}$.

an additive set function on D..

82
Theorem 3.14. For any non-additive measurefunction

on

$\nu$

$D_{u}$(or$D_{s}$

on$\mu$$\mathcal{X}$there exists a unique additive set) so that $\mu(A)=\nu(\tilde{A})$ for every$A\in \mathcal{X}$. Furthermore,$\mu$is the(signed) zeta transform of .
$\nu$

Corollary 3.15. For any non-additive measurefunction

$\nu$

on

$D_{u}$(or$D_{s}$

on$\mu$$\mathcal{X}$there exists a unique additive set) so that(C) $\int fd\mu=\int\tilde{f}d\nu$

for every measurable function f.
We shall call$\nu$

the Mobius transform of

on

$\mu$

$D_{p}$(or$D_{u}$) and denote it$\nu^{\mu}$.

The M\"obius transform is not always monotone. The next proposition shows the nec-

essary and sufficient condition for the Mobius transform to be monotone.
Proposition 3.16. A non-additive measureM\"obius transformation$\nu^{\mu}$

$\mu$

on

$\mathcal{X}$is totally monotone if and only if its

is monotone.

Since M\"obius transformation is additive, the monotonicity is equivalent to the positiveness of it.
The next two theorems show that the M\"obius transform can be extended to a$\sigma-$

additive (signed) measure.
Theorem 3.17. Any additive set function

$\nu$

on

$D_{s}$

has a unique -additive extension to the -algebra$\sigma$$\sigma$

Theorem 3.18. Let
set function$\nu^{\mu}$on$(X, \mathcal{X}, \mu)$$\sigma(\tilde{\mathcal{X}})$is$\sigma-$$\sigma(\tilde{\mathcal{X}})$

additive. If

generated by$\nu\in \mathcal{F}\mathcal{M}$$\mathcal{X}\sim$$in$, then it$2^{F\mathcal{M}_{0}^{1}}$.

be a non-additive measure space. There exists an additive

generated by$\tilde{\mathcal{X}}$in$2^{F\mathcal{M}_{0s}^{1}}$.

83
4Comparison
In this section we show that there exists abijection from the class of 0–1 non-additive

measures to the class of semifilters of $X$ .
Theorem 4.1. Let

be the class of 0-1 non-additive measures, $Sx$ be the class of$\mathcal{F}\mathcal{M}_{0}^{1}$

semifilters of X. There exists a bijection
We call the bijection$\varphi$$\varphi$:$\mathcal{F}\mathcal{M}_{0}^{1}arrow Sx$

in previous theorem amediator for representation.

The next theorem follows from the definition ofTheorem 4.2. Let.

$(X, \mathcal{X}, \mu)$$h_{f}$.

be a non-additive measure space, $(S_{X},S_{X}, m, H_{X})$ is an in-

terpreter representation, and $(hf, m)$ be the Choquet integral representation for $(f, \mu)$ , $f\in$$B^{+}$. Then we have$h_{f}( \nu):=\sup\{r|\nu\in H_{X}(\{x|f(x)\geq r\})\}$,

that is, $h_{f}=if$ .
As to the interpreter representation and representation with the M\"obius transform,

we have the next theorem.
Theorem 4.3. Let$\varphi$

$i$

be

$a$

interpreter from Ato $Sx$ ,$\tau$

be a tilde operator frombe the mediator for representation. Define a mapping$\{\theta\in S_{X}|A\in\theta\}$:$\mathcal{X}arrow 2^{S\chi}$to$\tilde{\mathcal{X}}$,by $H_{X}(A):=$.

(1) $Hx(A)=\varphi(\overline{A})$ for(2)$H_{X}$$\mathcal{X}$$i_{f}\mathrm{o}\varphi=\tau(f)$for$A\in \mathcal{X}$$f\in \mathcal{L}^{\infty+}$.
.

As we show above, the representing integrand are equivalent. On the other handconcerning the representing measure, in the interpreter representation the measure is

84
monotone and the uniqueness is not always true. In the representation with M\"obiustransform, the measure is not always monotone and the existence is unique.
It is not proved in [5] that there exists an additive set function on $D$ so that it represents

anon-additive measure and the Choquet integral. Using the interpreter representation

theorem, we can show the existence.

Theorem 4.4. For any non-additive measure

$\mu$

on

$\mathcal{X}$there exists a measure on $D$ sothat(C) $\int fd\mu=\int\tilde{f}d\nu$

for every$f\in \mathcal{L}^{\infty+}$.

Concerning belief function on the finite $X$ , the two theorems are perfectly the same.
We conclude this paper by showing the next table about the properties of three representation theorems.

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