数理解析研究所講究録
第 1819 巻 2012 年 65-70

65

CHOQUET INTEGRAL REPRESENTATION OF
COMONOTONICALLY ADDITIVE FUNCTIONALS

信州大学工学部 河邊淳*(Jun Kawabe)

Faculty of Engineering, Shinshu University

ABSTRACT. The Daniell-Stone type representation theorem of Greco leads us toan improvement of the Riesz type representation theorem of Sugeno, Narukawa,and Murofushi for comonotonically additive, monotone functionals.

1. INTRODUCTION

Let $X$ be a locally compact Hausdorff space. Let
denote the space of allnonnegative, continuous functions on $X$ with compact support and let
denote$X$the space of all nonnegative, continuous functions onvanishing at infinity. In [8],Sugeno et al. succeeded in proving an analogue of the Riesz typ integral representation theorem in nonadditive measure theory. More precisely, they gave a direct
proof of the assertion that every comonotonically additive, monotone functional on
can be represented as the Choquet integral with respect to a nonadditive
measure on $X$ with some regularity properties. Their theorem gives a functional
analytic characterization of the Choquet integrals and is inevitable in order to develop nonadditive measure theory based on the topology of the underlying spaces
on which measures are defined.
In this paper, we give an improvement of the above theorem with the help of
the Greco theorem [4], which is the most general Daniell-Stone type integral representation theorem for comonotonically additive, monotone functionals on function
spaces. By using the same approach, we also give a Riesz type integral representation
theorem for a bounded functional on.
$C_{00}^{+}(X)$$C_{0}^{+}(X)$$e$$C_{00}^{+}(X)$$C_{0}^{+}(X)$2010 Mathematics Subject Classification. Primary $28C05$ ; Secondary $28A12,28C15,28E10.$
Key words and phrases. nonadditive measure, comonotonic additivity, Choquet integral, Riesz
type integral representation theorem.
*Research supported by Grant-in-Aid for Scientific Research (C) No. 20540163, Japan Society
for the Promotion of Sciences (JSPS).

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2. NOTATION AND PRELIMINARIES
denote the family of all subsets of $X$ . For each
Let $X$ be a non-empty set and let
denote the set$A\subset X$ , letdenote the characteristic function of . Let andof all real numbers and the set of all nonnegative real numbers, respectively. Also letdenote the set of all extended real numbers and the set of all nonnegative
andextended real numbers, respectively. Let denote the set of all natural numbers.
$:= \min(f, g)$ . For, let $f\vee g$ $:= \max(f, g)$ andFor any functions $f,$ :$2^{X}$$A$$\mathbb{R}$$\mathbb{R}^{+}$$\chi_{A}$$\frac{\mathfrak{l}}{\mathbb{R}^{I}}$$\overline{\mathbb{R}}$$\mathbb{N}$$f\wedge g$$Xarrow\overline{\mathbb{R}}$$g$any bounded , let$f$$\Vert f\Vert_{\infty}$$:= \sup_{x\in X}|f(x)|.$Definition 1. set function :and $\mu(A)\leq\mu(B)$ whenever$2^{X}arrow\overline{\mathbb{R}}^{+}$

$A$

$\mu$

$\mu(\emptyset)=0$

is called a nonadditive measure on $X$ if$A\subset B.$

be a function. Since
Let be a nonadditive measure on $X$ and let :the function $t\in \mathbb{R}^{+}\mapsto\mu(\{f>t\})$ is non-increasing, it is Lebesgue integrable on. Therefore, the following formalization is well-defined; see [2] and [7].
$X arrow\frac{\mathfrak{l}}{\mathbb{R}^{I}}$

$f$

$\mu$

$\mathbb{R}^{+}$Definition 2. Let be a nonadditive measure on $X$ . The Choquet integral of awith respect to is defined bynonnegative function :$\mu$$Xarrow\overline{\mathbb{R}}^{+}$

$f$

$\mu$

(C) $\int_{X}fd\mu:=\int_{0}^{\infty}\mu(\{f>t\})dt,$

where the right hand side of the above equation is the usual Lebesgue integral.

Remark 1. For any nonadditive measure on $X$ and any function :are equal, sinceand
the two Lebesgue integrals$\mu(\{f\geq t\})\geq\mu(\{f>t\})\geq\mu(f\geq t+\epsilon\})$ for every
and $0\leq t<\infty$ . This fact
will be used implicitly in this paper.
$f$

$\mu$

$\int_{0}^{\infty}\mu(\{f>t\})dt$$X arrow\frac{1}{\mathbb{R}^{I}},$$\int_{0}^{\infty}\mu(\{f\geq t\})dt$$\epsilon>0$

See [3], [6], and [9] for more information on nonadditive measures and Choquetintegrals.
For the reader’s convenience, we introduce the Greco theorem [4, Proposition 2.2],
which is the most general Choquet integral representation theorem for comonotonically additive, monotone, extended real-valued functionals. Recall that two funcare comonotonic and is written by $f\sim g$ if, for every $x’\in X,$
tions $f,$$x,$$g:Xarrow\overline{\mathbb{R}}$$f(x)<f(x’)$ implies $g(x)\leq g(x’)$ .

Theorem 1 (The Greco theorem). Let. Assume that satisfies$Xarrow\overline{\mathbb{R}}$$\mathcal{F}$be a non-empty family of functions$\mathcal{F}$(i)(ii) $f\geq 0$ for every$0\in \mathcal{F},$$f\in \mathcal{F}$(nonnegativity), and

$f$

:

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(iii) ifandcondition).
$f\in \mathcal{F}$$c\in \mathbb{R}^{+}$

Assume that a functional, then $cf,$$I:\mathcal{F}arrow\overline{\mathbb{R}}$$f\wedge c,$$f-f\wedge c=(f-c)^{+}\in \mathcal{F}$(the Stonesatisfies(iv) $I(0)=0,$(v) if $f,$$g\in \mathcal{F}$(vi) if $f,$$g\in \mathcal{F},$and $f\leq g$ , then $I(f)\leq I(g)$ (monotonicity),, andthen $I(f+g)=I(f)+I(g)$ (comonotonic$f+g\in \mathcal{F}$$f\sim g_{f}$additivity),(vii) $\lim_{aarrow+0}I(f-f\wedge a)=I(f)$ for every(viii) $\lim_{barrow\infty}I(f\wedge b)=I(f)$ for every

For each $A\subset X$ , define the set functions$f\in \mathcal{F}$, and$f\in \mathcal{F}.$$\alpha,$$\beta$:$2^{X} arrow\frac{\mathfrak{l}}{\mathbb{R}^{1}}$by$\alpha(A):=\sup\{I(f):f\in \mathcal{F}, f\leq\chi_{A}\},$$\beta(A):=\inf\{I(f):f\in \mathcal{F}, \chi_{A}\leq f\},$where let$inf\emptyset$$:=\infty.$(1) The set functions

are nonadditive measures on $X$ with
(2) For any nonadditive measure on $X$ , the following two conditions are equivalent:$\alpha$and$\beta$$\alpha\leq\beta.$$\lambda$(a)$\alpha\leq\lambda\leq\beta.$(b) $I(f)=( C)\int_{X}fd\lambda$ for every$f\in \mathcal{F}.$

Remark 2. The functional given in Theorem 1 is nonnegative, that is, $I(f)\geq 0$
for every, and positively homogeneous, that is, $I(cf)=cI(f)$ for every
andSee, for instance, [3, page 159] and [5, Proposition 4.2].
$I$$f\in \mathcal{F}$$f\in \mathcal{F}$$c\in\overline{\mathbb{R}}^{+}$

3. RIESZ TYPE INTEGRAL REPRESENTATION THEOREMSIn this section, we give an improvement of the Sugeno-Narukawa-Murofushi theorem [8, Theorcm 3.7]. This can be done by the effective usc of the Greco theorem
and the following technical lemma.

Lemma 1. Let$\mathcal{F}$and I satisfy the same hypotheses as Theorem 1.

(1) Assume that, for any, there is asuch that
and$I(g)<\infty$ $(in$ particular, $1\in \mathcal{F} and I(1)<\infty$ ). Then, condition (vii)ofTheorem 1 holds.

(2) Assume that everyholds.

$f\in \mathcal{F}$$f\in \mathcal{F}$$g\in \mathcal{F}$$\chi_{\{f>0\}}\leq g$is bounded. Then, condition (viii) of Theorem 1

68is bounded. Also assume that I is bounded, thatfor every, there is a constant $M>0$ such thatThen, conditions (vii) and (viii) of Theorem 1 hold.

(3) Assume that every$f\in \mathcal{F}$$f\in \mathcal{F}.$$I(f)\leq M\Vert f\Vert_{\infty}$$is$

From this point forwards, $X$ is a locally compact Hausdorff space. For any realvalued function on $X$ , let $S(f)$ denote the support of , which is defined by the$f$$f$closure of $\{f\neq 0\}.$The following regularity properties give a tool to approximate general sets bymore tractable sets such as open and compact sets. They are still important in

nonadditive measure theory.

Definition 3. Let$\mu$be a nonadditive measure on $X.$

is said to be outer regular if, for every subset of $X,$ $\mu(A)=iryf\{\mu(G)$ :$A\subset G,$
is open}.
(2)
is said to be quasi outer regular if, for every compact subset $K$ of $X,$
is open$\mu(K)=\inf\{\mu(G)$ : $K\subset G,$(3) is said to be inner Radon if, for every subset of $X,$ $\mu(A)=\sup\{\mu(K)$ :(1)$A$$\mu$$G$$\mu$$G$$\}.$$A$$\mu$

is compact}.
is said to be quasi inner Radon if, for every open subset$K\subset A,$ $K$(4)$\mu$$\sup${$\mu(K)$$G$of $X,$ $\mu(G)=$: $K\subset G,$ $K$ is compact}.

The following theorem is an improvement of [8, Theorem 3.7] and it has essentiallybeen derived from the Greco theorem.
Theorem 2. Let a functional$I$:satisfy the following conditions:$C_{00}^{+}(X)arrow \mathbb{R}$(i) if $f,g\in C_{00}^{+}(X)$ and $f\leq g$ , then $I(f)\leq I(g)$ (monotonicity), and(ii) if $f,$ $g\in C_{00}^{+}(X)$ and $f\sim g$ , then $I(f+g)=I(f)+I(g)$ (comonotonicadditivity).
For each $A\subset X$ , define the set functions$\alpha,$$\beta,$$\gamma$:$2^{X}arrow\overline{\mathbb{R}}^{+}$by$\alpha(A):=\sup\{I(f):f\in C_{00}^{+}(X), f\leq\chi_{A}\},$$\beta(A) :=\inf\{I(f) :f\in C_{00}^{+}(X), \chi_{A}\leq f\},$$\gamma(A):=\sup\{I(f):f\in C_{00}^{+}(X), 0\leq f\leq 1, S(f)\subset A\},$

where let$inf\emptyset$$:=\infty$, and their regularizations$\alpha^{*}(A)$$\beta^{**}(A)$$\gamma^{*}(A)$: $A\subset G,$$:= \inf\{\alpha(G)$$:= \sup${$\beta(K)$$:= \inf\{\gamma(G)$$\alpha^{*},$$G$$\beta^{*},$$\gamma^{*}:2^{X}arrow\frac{\mathfrak{l}}{\mathbb{R}^{I}}$is open$\},$: $K\subset A,$ $K$ is compact},: $A\subset G,$$G$

is open$\}.$by69(1) The set functions$\alpha,$$\beta,$$\gamma,$$\alpha^{*},$, andare nonadditive measures on $X.$
on $X$ , the following two conditions are equiv-$\beta^{**}$(2) For any nonadditive measure$\lambda$$\gamma^{*}$

alent:(a)$\alpha\leq\lambda\leq\beta.$(b) $I(f)=( C)\int_{X}fd\lambda$ for every $f\in C_{00}^{+}(X)$ .
(3) $\gamma^{*}(K)=\beta(K)<\infty$ for every compact subset $K$ of $X.$

(4)$\gamma^{*}$is quasi inner Radon and outer regular.

(5)
is inner Radon and quasi outer regular.
(6) $\beta^{**}(G)=\gamma(G)$ for every open subsetof $X.$
(7) The defined nonadditive measures are compamble, that is,, so that any of them is a representing measure of$\beta^{**}$$G$$\alpha=\gamma\leq\beta^{**}\leq$$\alpha^{*}=\gamma^{*}\leq\beta$$I.$

Remark 3. Define the functional :by $I(f)$for every. Then satisfies (i) and (ii) of Theorem 2, but it is not bounded. So,Theorem 2 does not follow from (3) of Lemma 1.
$I$$C_{00}^{+}(\mathbb{R})arrow \mathbb{R}$$:= \int_{-\infty}^{\infty}f(t)dt$$I$$f\in C_{00}^{+}(\mathbb{R})$From Theorem 2 and Lemma 1, we can derive a representation theorem forbounded, comonotonically additive, monotone functionals on.
$C_{0}^{+}(X)$Theorem 3. Let a functional$I$:satisfy$C_{0}^{+}(X)arrow \mathbb{R}$(i) if $f,$ $g\in C_{0}^{+}(X)$ and $f\leq g$ , then $I(f)\leq I(g)$ (monotonicity),(ii) if $f,$ $g\in C_{0}^{+}(X)$ and $f\sim g$ , then $I(f+g)=I(f)+I(g)$(comonotonicadditivity), and(iii) there is a constant $M>0$ such that

$I(f)\leq M\Vert f\Vert_{\infty}$

for every$f\in C_{0}^{+}(X)$(boundedness).

For each $A\subset X$ , define the set functions$\alpha,$$\beta,$$\gamma$:$2^{X}arrow\overline{\mathbb{R}}^{+}$

by$\alpha(A):=\sup\{I(f):f\in C_{0}^{+}(X), f\leq\chi_{A}\},$$\beta(A):=\inf\{I(f):f\in C_{0}^{+}(X), \chi_{A}\leq f\},$$\gamma(A):=\sup\{I(f):f\in C_{0}^{+}(X), 0\leq f\leq 1, S(f)\subset A\},$

where let$inf\emptyset$$:=\infty$, and their regularizations$\alpha^{*}(A)$$\beta^{**}(A)$$\gamma^{*}(A)$(1) The set functions: $A\subset G,$$:= \inf\{\alpha(G)$$:= \sup${$\beta(K)$$:=i_{YJ}f\{\gamma(G)$$\beta^{**},$$\alpha^{*},$$\gamma^{*}:2^{X}arrow\overline{\mathbb{R}}^{+}$

is open$G$by$\},$: $K\subset A,$ $K$ is compact},: $A\subset G,$$G$

is open$\}.$

are nonadditive measures on $X$
and they satisfy properties (3) $-(7)$ in Theorem 2.
$\alpha,$$\beta,$$\gamma,$$\alpha^{*},$$\beta^{**}$, and$\gamma^{*}$

70(2) For any nonadditive measure$\lambda$

on $X$ , the following two conditions are equiv-

alent:(a)$\alpha\leq\lambda\leq\beta.$(b) $I(f)=( C)\int_{X}fd\lambda$ for every $f\in C_{0}^{+}(X)$ .
(3) $\alpha(X)=\gamma(X)=\alpha^{*}(X)=\gamma^{*}(X)<\infty.$(4) Let$\lambda$

be a nonadditive measure on $X$ with $\lambda(X)<\infty$ . Define the functional
by $I(f):=( C)\int_{X}fd\lambda$ for every $f\in C_{0}^{+}(X)$ . Then I satisfies$I:C_{0}^{+}(X)arrow \mathbb{R}$conditions $(i)-(iii)$ .
4. CONCLUSION
In this paper, we gave an improvement of the Riesz type integral representation
theorem of Sugeno, Narukawa, and Murofushi by the help of the Daniell-Stone type
integral representation theorem of Greco. By using the same approach, we also gave.
a Riesz type integral representation theorem for a bounded functional on$C_{0}^{+}(X)$Our approach will lead us to various Riesz type integral representation theorems ona wide variety of function spaces and sequence spaces.
REFERENCES[1] Berberian, S.K.: Measure and Integration. Macmillan, New York (1965)[2] Choquet, G.: Theory of capacities. Ann. Inst. Fourier Grenoble 5, 131-295 (1953-54)[3] Denneberg, D.: Non-Additive Measure and Integral. 2nd ed. Kluwer Academic Publishers,Dordrecht (1997)[4] Greco, G.H.: Sulla rappresentazione di funzionali mediante integrali. Rend. Sem. Mat. Univ.
Padova 66, 21-42 (1982)[5] Namkawa, Y., Murofushi, T., Sugeno, M.: Regular fuzzy measure and representation ofcomonotonically additive functional. Fuzzy Sets and Systems 112, 177-186 (2000)[6] Pap, E.: Null-Additive Set Functions. Kluwer Academic Publishers, Dordrecht (1995)[7] Schmeidler, D.: Integral representation without additivity. Proc. Amer. Math. Soc. 97, 255-261(1986)[8] Sugeno, M., Namkawa, Y., Murofushi, T.: Choquet integral and fuzzy measures on locallycompact space. Fuzzy Sets and Systems 99, 205-211 (1998)[9] Wang, Z., Klir, G.J.: Generalized Measure Theory. Springer, Berlin Heidelberg, New York(2009)

DEPARTMENT OF MATHEMATICSFACULTY OF ENGINEERINGSHINSHU UNIVERSITY
4-17-1 WAKASATO, NAGANO 380-8553, JAPAN$E$-mail address: jkawabe@shinshu-u.ac.jp