Narukawa, Yasuo

Kyoto University Research Information Repository

TitleInner and Outer Choquet integral representation (AdvancedStudy of Applied Functional Analysis and InformationSciences)Author(s)Narukawa, YasuoCitation数理解析研究所講究録 (2005), 1452: 184-194Issue Date2005-10URL http://hdl.handle.net/2433/47769RightType Departmental Bulletin PaperTextversionpublisherKyoto University184Inner and Outer Choquet integral representation(Yasuo Narukawa)Toho Gakuen1 IntroductionThe Choquet integral is commonly used as the integral for a non-additive set function, becausethe Choquet integral coincides with expectation in probability theory when the set function isprobability measure. The most important properties of the Choquet integral are comonotonicadditivity and monotonieity (for short we call $\mathrm{c}.\mathrm{m}.$).Considering the topology, various regularities are proposed and studied [4, 5, 12, 17, 16, 8,11, 14], especially, [16] propose their regularity in relation to Choquet integral, that is, a $\mathrm{c}\mathrm{m}$ .functional on the class of continuous functions with compact support is represented by Choquetintegral with respect to an $0$-regular non-additive measure.In this paper, we assume the universal set $X$ to be a locally compact Hausdorff space.Wesay that the representation using $0$ -regular non-additive measure defined by the supremum ofthe functional is an outer representation. On the other hand, the representation that uses theinfinimum of the functional, that is called an inner representation, can be considered. In thispaper we study $\mathrm{i}$-regular non-additive measure and inner representation. We make clarify thedifference between $\mathrm{o}$ regularity and $\mathrm{i}$ regularity or betw een an outer representation and an innerrepresentation. The structure of this paper is as folows: In section 2 we present basic definition$\mathrm{s}$1452 2005 184-194185and properties without topological assumption of non-additive measure and Choquet integralwith respect to a non-additive measure as a preliminaries. In Section 3 we assume that theuniversal space $X$ is locally compact space. We define $0$-regular and $\mathrm{i}$-regular non-additivemeasure and show some properties.In section 4, we study the representation of a $\mathrm{c}\mathrm{m}$ . functional. We show that the innerrepresentation is possible, although the method is somewhat different from the outer one and theproof is rather complex. The induced $\mathrm{i}$-regular non-additive measure coincides with o-regularone on the class of compact sets.In section 5, we finish with some concluding remark.2 Non-additive measure and Choquet integralIn this section, we present some basic definition and properties of non-additive measure theory.$X$ denotes the universal set and $B$ denotes its 0- algebra. No topological assumption is neededin this section.In this paper, we distinguish the term “fuzzy measure from “non-additive measur\"e. Sugeno’$\mathrm{s}$original axioms [15] for a fuzzy measure has some continuity. On the other hand, some authorsdefine a fuzzy measure, that is monotone set function vanishing at 0, and is not assumed anycontinuity. In order to avoid confusion, in this paPer, according to Denneberg’s monograph [2],we say it a non-additive measure.Definition 2,1. A non-additive measure $\mu$ is an extended real valued set function, $\mu$ :$tB$ $arrow\overline{R^{+}}$with the following properties;$\mu(\emptyset)=0$, and $\mathrm{j}\mathrm{j}(\mathrm{A})\leq\mu(B)$ whenever $A\subset B$ , $A,B\in B$ where $\overline{R^{+}}=[0,\infty]$ is the set of extendednonnegative real numbers. In this paper we assume that $\mu$ is finite, that is, $\mu(X)<\infty$ .The class of non-negative measurable functions is denoted by $M^{+}$ .186Definition 2,2. [1, 2] Let $\mu$ be a non-additive measure on (X, B).1. The Choquet integral of f $\in M^{+}$ with respect to $\mu$ is defined by(C) $ffd \mu=\int_{0}^{\infty}\mu_{f}(r)dr$,when $\mu f(r)=\mu(\{x|f(x) \geq r\})$ .Definition 2.3. [3] Let $f$, $g\in M^{+}$ . We say that $f$ and $g$ are comonotonic if $f(x)$ $<f(x’)\Rightarrow$$g(x)$ $\leq g(x’)$ for $x$ , $x’\in X$ . We denote $f\sim g$, when $f$ and $g$ are comonotonic.The Choquet integral of $f\in M$ with respect to a non-additive measure has the next basicproperties.Theorem 2.3. [2] Let f, g $\in M^{+}$ .1. (Monotonicity) Iff $\leq g$, then(C) $\int fd\mu\leq(C)\int gd\mu$2. (Comonotonic additivity) Iff $\sim g$, then(C)$) \int(f+g)d\mu=(C)$ $I^{fd\mu+}(\backslash c)\prime gd\mu$ .3 Properties of $\mathrm{i}$-regular non-additive measureIn this section we show some properties of 0- regular non-additive measure and $\mathrm{i}$-regular non-additive measure. The properties of $0$-regular non-additive measure are shown in [6, 7, 9, 10, 8,11].In the following we assume that $X$ is a locally compact Hausdorff space, $lB$ the class ofBorelsubsets of $X$ , $O$ the class of open subsets of $X$ and $C$ the class of compact subsets of $X$ .$C^{+}(X)$ denotes the class ofnon-negative continuous functions with compact support $\mathrm{a}\mathrm{n}\mathrm{d}C_{1}^{+}(X)$denotes the subclass of $C^{+}(X)$ with $0\leq f\leq 1$ for $f\in C_{1}^{+}(X)$ .187Definition 3.1. Let $\mu$ be a non-additive measure on the measurable space $(X, B)$ .1. $\mu$ is said to be $0$ -continuous from below if $O_{n}\uparrow O\Rightarrow\mu(O_{n})\uparrow\mu(O)$ where $n=1,2,3$ , $\cdots$and both $O_{n}$ and $O$ are open sets .2. $\mu$ is said to be $c$-continuous from above if $C_{n}\downarrow C\Rightarrow\mu(C_{n})\downarrow\mu(C)$ where n $=1,$ 2, \cdots andboth $C_{n}$ and C are compact sets.First, we define the regular non-additive measures.Definition 3.2. Let $\mu$ be a non-additive measure on measurable space (X, B). $\mu$ is said to beinner regular if$\mu(B)=\sup\{\mu(C)|C\in C,C\subset B\}$for all B $\in B$ .Inner regular non-additive measure is called $\mathrm{i}-regular$ if$\mu(C)=\inf\{\mu(O)|O\in O, C\subset O\}$for all $C\in C$.$\mu$ is said to be outer regular if$\mu(B)=\inf\{\mu(O)|O\in O, O\supset B\}$ for all $B\in B$ .Outer regular non-additive measure is called $\mathit{0}$ -regular if$\mu(O)=\sup\{\mu(C)|C\in O,C\subset O\}$for all $O\in O$ .The next lemmas follow from the definition immediately.Lemma 3.3. Let $\mu i$ be an $\mathrm{i}$-regular non-additive measure and $\mu_{\mathit{0}}$ be an o-regular non-additivemeasure. $\mu_{i}(O)=\mu_{\mathit{0}}(O)$ for $0\in O$ if and only if$\mu_{i}(C)=\mu_{\mathit{0}}(C)$ for C $\in C$.Lemma 3.1. Let $\mu_{i}$ be an $\mathrm{i}$-regular non-additive measure and $\mu_{\mathit{0}}$ be an $0$ -regular non-additivemeasure such that $\mu_{i}(C)=\mu_{\mathit{0}}(C)$ for C $\in C$. Then we have $\mu_{i}(A)\leq\mu_{\mathit{0}}(A)$ for all A $\in B$ .The next two results follow from the definition.Proposition 3.5, [6] Let $\mu$ be a o- regular non-additive measure.1. $\mu$ is $0$ -continuous from below1882. pt is $c$ -continuousfrom above.Remark Proposition 3.5 (2) is true when non-additive measure are outer regular non-additivemeasures.We have the next lemma from the definition and Ulysohn’s lemma, although it is still openwhether the similar continuity to the lemma above holds in the case of $\mathrm{i}$-regular non-additivemeasure.Proposition 3.6. Let $\mu$ be an $\mathrm{i}$-regular non-additive measure. For every compact set C $\in C$ thereexists a sequence $\{O_{n}\}$ ofopen sets such that $\mu(O_{n})\downarrow\mu(C)$ .In relation to the Choquet integral, we have the next propositions.Proposition 3.7. [6] Let $\mu_{1}$ and $\mu_{2}$ be $c$-continuous from above.Iffor all $f \in K(C)\int fd\mu_{1}=(C)\int fd\mu_{2}$ , then $\mu 1(C)=\mu 2(C)$ for all $C\in C$.In the case of $\mathrm{i}$-regular, we have the next similar result.Proposition 3.8. Let $\mu$ be an $\mathrm{i}$-regular non-additive measure.Iffor all $f \in K(C)\int fd\mu_{1}=(C)\int fd\mu_{2}$ , then $\mu_{1}(C)=\mu 2(C)$ for all $C\in C$.It follows from Proposition 3.6 that we have$\mu_{i}(\{x|f^{n}(x)\geq\alpha\})\downarrow\mu_{i}(\{x|1_{C}(x)\geq\alpha\})$as $narrow\infty$ for $0<\alpha\leq 1$ and $\mathrm{i}=1,2$ .The next theorem is the main theorem of this chapter.Theorem 3,9. Let $\mu_{i}$ be $\mathrm{i}$-regular non-additive measure and $\mu_{\mathit{0}}$ be $0$ -regular non-additive mea-sures. Iffor all $f \in C^{+}(C)\int fd\mu_{i}=(C)\int fd\mu_{\mathit{0}}$ , then1. $\mu_{i}(C)=\mu_{\mathit{0}}(C)$ for all $C\in C$,1882. $\mathrm{t}\mathrm{n}(\mathrm{O})=\mu_{\mathit{0}}(O)$ for all $C\in O$,3. $\mu_{i}(A)\leq\mu_{\mathit{0}}(A)$ for all $A$ $\in B$.4 Representation theoremsThe Choquet integral is a comonotonically additive and monotone functional. Conversely, weconsider a $\mathrm{c}$ . $\mathrm{m}$ . functional on the class of continuous functions with compact support.Definition 4.1. Let I be a real valuedfunctional on $C^{+}(X)$ .We say that I is comonotonically additive iff $f\sim g\Rightarrow I(f+g)=I(f)+I(g)$ for $f,g\in$$C^{+}(X)$ , and that I is monotone iff$f\leq g\Rightarrow I(f)\leq I(g)$ for $f,g\in C^{+}(X)$ .$Ifa$ functional I is comonotonically additive andmonotone, we say that I is a $\mathrm{c}\mathrm{m}$ . functional.First, we review the outer representation.Lemma 4.2. [16J Let I be a c.m. functional on $C^{+}$ .We put $f_{I}(O)= \sup${ $I(f)|f\in C_{1}^{+}$ , supp(f)\subset O} for $O\in O$and $\mu_{I}^{o}(B)=\inf\{f_{I}(O)|O\in O, O\supset B\}$ for $B\in B$ . Then $\mu_{I}^{o}$ is an outer regular non-additivemeasure.We shall say that this outer regular non-additive measure $\mu_{I}^{o}$ is an outer non-additive measureinduced by a cm. functional I.Proposition 4.3. [16] Let $\mu_{I}^{o}$ be an outer non-additive measure induced by a cm. functional I.1. If $f\in C_{7}^{+}A\subset\{x|f(x)\geq 1\}$ , $andA\in B$ , then $\mathrm{I}(\mathrm{g})$ $\leq I(f)$2. IfC is a compact set, then $\mu l(C)<\infty$.3. If 0 is an open set, then $\mu I(O)=\sup\{\mu I(C)|C\in C,$C $\subset O\}$ .1aoTherefore the non-additive measure induced by a c.m- functional is o-regular.From the above lemma, we obtain the following theorem [16].Theorem 4.4. (Outer representation theorem) [16]Let I be a $\mathrm{c}\mathrm{m}$. functional on $C^{+}$ . If $lP_{I}$ is an outer regular non-additive measure induced by$I$, then we have $I(f)=(C) \int fd\mu_{I}^{O}$for all $f\in C^{+}(X)$ .In the following, we present some preliminary results for inner representation. The proofsare in $[16, 6]$ .Let $\mathit{0}_{n,k}=\{x|f(x)>\frac{k-1}{n}\}$ : $CU|k= \{x|f(x)\geq\frac{k}{n}\}$ where $f\in C^{+}$ , $k$ and $n$ is a positive integerand $1\leq k\leq n$ . Then for all $n$ and $k$ , $C_{n,k}$ is a compact set, $O_{n,k}$ is an open set, and $O_{n_{7}k+1}\subset$$C_{n,k}\subset O_{n,k}\subset supp(f)$ . Since $X$ is a locally compact Hausdorff space, for all $n,k$ there exists$f_{n,k}\in C_{1}^{+}$ such that $f_{n,k}(x)=1$ when $x$ $\in C_{n,k}$ and $supp(f_{n,k})\subset \mathit{0}_{n,k}$ .The functions $f_{n,k}$ have the following properties.Lemma 4.5. fl6f1. 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 fnj arecomonotonic.2. For all positive integer n and k such that $1\leq k\leq n$,$f_{n,1}+f_{n,2}+\cdots+f_{n,k}$ and $f_{n,k+1}+f_{n,k+2}+\cdots+f_{n,n}$ are comonotonic.Lemma 4.6. [16] Define $f_{n}\in C_{1}^{+}for$ n $=1,2$ , \cdots by $f_{n}= \sum_{k=1}^{n}\frac{1}{n}f_{n,k}$ .1. there exists $F\in c_{1}^{+}$ which satisfies the following conditions.(a) $0 \leq|f-f_{n}|\leq\frac{1}{n}F$(b) $x\in$ supp(f) $\Rightarrow F(x)=1$191(c) $f_{n}\sim_{c}F$ for all $n$2. We have $||f-f_{n}|| \leq\frac{1}{n}$ , where ||.|| is the $\sup$ norm.3. Suppose that $\frac{k}{n}\leq f(x)<\frac{k+1}{n}$ , then we have $\frac{k}{n}\leq f_{n}(x)$ $\leq\frac{k+1}{n}$4. $\lim_{narrow\infty}I(f_{n})=I(f)$Now we can proceed the inner representation. The definition of a non-additive measure is alittle bit different from outer representation. The proof is obvious from the definition.Lemma 4,7. Let I be a $\mathrm{c}\mathrm{m}$ . functional on $C^{+}$ .We put $\mu_{I}^{i}(C)=\inf\{I(f)|f\in C_{1}^{+},1_{C}\leq f\}$ for $C\in C$ and $\mu_{I}^{i}(B)=\sup\{\mu_{I}^{i}(C)|C\in C,C\subset B\}$for $B\in B$ . Then $\mu_{I}^{i}$ is an inner regular non-additive measure.We shall say that the non-additive $\mu_{I}^{i}$ is an inner regular non-additive measure induced by a$\mathrm{c}\mathrm{m}$ . functional $I$ .Proposition 4.8.Let $\mu_{I}^{i}$ be an inner non-additive measure induced by a $\mathrm{c}\mathrm{m}$ . functional I. Suppose that$f\in C_{1}^{+},A\supset sup(f)$ , and $A\in I\mathit{3}$. Then we have $\mu_{I}^{i}(A)\geq I(f)$ .Lemma 4.9. Let $\mu_{I}^{i}$ be an inner non-additive measure induced by a c.m. functional I. Supposethat f $\in C_{1)}^{+}O\subset supp(f)$ , and O $\in O$. Then we have $\mu_{I}^{i}(O)\leq I(f)$ .Proposition 4.8. Let $\mu_{I}^{i}$ be an inner non-additive measure induced by a $c.m$. functional I. Then$\mu_{J}^{i}$ is $\mathrm{i}$-regulan that is, $\mu_{I}^{i}(C)=\inf\{\mu_{I}^{i}(O)|O\in O,O\supset C\}$ .Applying the above mentioned lemmas we can prove the inner representation theorem.Theorem 4.11182Let I be a $\mathrm{c}\mathrm{m}$. functional on $C^{+}$ . If $\mu_{I}^{i}$ is an $\mathrm{i}$-regular non-additive measure induced by $I$,then we have$I(f)=(C) \oint fd\mu_{I}^{i}$for all $f\in C_{1}^{+}$ .The next theorem follows from Theorem 3.9, Theorem 4,4 and Theorem 4.11.Corollary 4.12. Let I be a cm. functional on $C^{+}and$ $f_{l}$ (resp. $\mu_{I}^{i}$ ) is an o- regular (resp. an$\mathrm{i}$-regular non-additive measure induced by I,1. $\mu_{I}^{i}(C)=\mu_{I}^{o}(C)$ for all $C\in C$,2. $\mu_{I}^{i}(O)=\mu_{I}^{o}(O)$ for all $C\in O$,3. $\mu_{i}(A)\leq\mu_{\mathit{0}}(A)$ for all $A\in B$.5 ConclusionsWe have studied the properties of an $i$-regular non-additive measure and shown that an innerrepresentation is possible. We showed that an $0$-regular non-additive measure and an i-regularnon-additive measure induced by a $\mathrm{c}\mathrm{m}$ . functional takes same value on the class of compactsets and the class of open sets. It will be future works whether they coincides on the class ofBorel sets and under what condition they coincides if they are not same on the class of Borelsets.References[1] G.Choquet . (1955), Theory of capacities . Ann. Inst. Fourier, Grenoble. 5, 131-295193[2] D. Denneberg, (1994) Non Additive Measure and Integral, Dordorecht:Kluwer AcademicPublishers.[3] C. Dellacherie, (1971) Quelques commentaires sur les prolongements de capacit\’es,Siminaire de Probabilites 1969/1970, Strasbourg, Lecture Notes in Mathematics, 191(Springer) 77- 81,[4] A. B. Ji, (1997), Fuzzy measure on locally compact space, The Journal of Fuzzy Mathe-matics, Vol. 5 No 4., 989-995.[5] J. Li, M. Yasuda, (2004), Lusin’s theorem on fuzzy measure spaces, Fuzzy sets and systemsVolume 146, Issue 1, pp. 121-133.[6] Y. Narukawa, T. Murofushi, M. Sugeno, (2000), Regular fuzzy measure and representa-tion of comonotonically additive functionals, Fuzzy Sets and Systems 112,(2), 177-186,[7] Y. Narukawa, T. Murofushi, M. Sugeno, (2001), Boundedness and Symmetry ofComono-tonically Additive Functionals, Fuzzy Sets and Systems 118, (3), 539-545.[8] Y. Narukawa, T. Murofushi,(2004) Regular Non-additive measure and Choquet integral,Fuzzy Sets and Systems, vol. 143, no. 3, pp. 487-492.[9] Y. Narukawa, T. Murofushi and M. Sugeno,(2001), Extension and representation ofcomonotonically additive functionals, Fuzzy Sets and System s, Vol. 121 no. 2pp. 217-226[10] Y. Narukawa and T. Murofushi (2003), Conditions for Choquet integral representation ofthe comonotonically additive and monotone functional, J. Math. Anal. Appl. vol. 282, no.1, pp. 201-211194[11] Y. Narukawa and T. Murofushi, (2004) Choquet integral with respect to a regular non-additive measures, Proc. 2004 IEEE Int. Conf. Fuzzy Systems (FUZZ-IEEE 2004), 517-521.[12] E. Pap, (1995), Regular null additive monotone set functions, Univ. u Novom Sadu Zh.rad. Prorod.-Mat. Fak set mat 25, 2, 93-101[13] E. Pap, (1995) Null-Additive Set Functions, Dordorechet: Kluwer Academic Publishers.[14] J. Song, J. Li, (2003), Regularity of null additive fuzzy measure on metric spaces. Inter-national Journal ofGeneral Systems, 32 (3) 271-279.[15] M. Sugeno, (1974) Theory fuzzy integrals and its applications, Doctoral Thesis, TokyoInstitute of Technology.[16] M. Sugeno, Y. Narukawa and T. Murofushi,(1998), Choquet integral and fuzzy measureson locally compact space, Fuzzy Sets and Systems, 99 (2) 205-211.[17] J. Wu, C. Wu,(2001) Fuzzy regular measures on topological spaces. Fuzzy sets andSystems, 119,529-533.

Inner and Outer Choquet integral representation (Advanced Study of Applied Functional Analysis and Information Sciences)

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Inner and Outer Choquet integral representation (Advanced Study of Applied Functional Analysis and Information Sciences)

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