Murofushi, Toshiaki

Narukawa, Yasuo

Kyoto University Research Information Repository

Title On regular non-additive measures (Advanced Topics ofInformation Science and Functional Analysis)Author(s)Narukawa, Yasuo; Murofushi, ToshiakiCitation数理解析研究所講究録 (2004), 1396: 100-111Issue Date2004-10URL http://hdl.handle.net/2433/25972RightType Departmental Bulletin PaperTextversionpublisherKyoto University100On regular non-additive measures( )(Yasuo NARUKAWA)Toho Gakuen ,(Toshiaki MUROFUSHI)Dept. Comp. Intell. & Syst. $\mathrm{S}\mathrm{c}\mathrm{L}$ , Tokyo Inst. Tech1 IntroductionNon-additive set functions on measurable space is used in economics, decision theoryand artificial intelligence, called by various name, such as cooperative game , capacity orfuzzy measure. In this paper, we distinguish the term ” fuzzy measures from “non-additivemeasur\"e. Sugeno’s original axioms [15] for a fuzzy measure has some continuity. On theother hand, some authors define a fuzzy measure, that is monotone set function vanishingat 0, and is not assumed any continuity. In order to avoid confusion, in this paper, wesay that monotone set function vanishing at G) is non-additive measure.Generally, considering an infinite set, if nothing is assumed, it is too general and issometimes inconvenient. Then we assume the universal set $X$ to be a locally compactHausdorff space. Considering the topology, various regularities are proposed [12, 17, 16].In this paper, we arrange the various regularities and clarify their correlation. Weconsider the relation among the regularities and the Choquet integral with respect to aregular non-additive measure and introduce some new results about the approximation1396 2004 100-111101of Choquet integral of a integrable function.The structure of this paper is as follows: In section 2 we present some basic definitionand properties without topological assumption of non-additive measure and Choquet in-tegral with respect to a non-additive measure as a preliminaries. In Section 3, we assumethat the universal space $X$ is locally compact space. We introduce some regularities ofnon-additive measure and show some properties. In Section 4, we show some proper-ties of the Choquet integral with respect to a regular non-additive measure. We showsome representation theorem of functionals on the class of continuous function with com-pact support and approximation theorem of Choquet integral of integrable functions. InSection 5, we finish with some concluding remark.2 PreliminariesIn this section, we present some basic definition and properties of non-additive measuretheory. $X$ denotes the universal set and $B$ its a-algebra. No topological assumption isneeded in this section.Definition 2.1. [2] A non-additive measure $\mu$ is an extended real valued set function,$\mu$ : $Barrow\overline{R^{+}}$ with the following properties: (1) $\mu(\emptyset)=0$ , (2) $\mu(A)\leq\mu(B)$ whenever$A\subset B,$ $A$ , $B\in B,$ where $\overline{R^{+}}=$ $[0, \infty]$ is the set of extended nonnegative real numbers. Inthis paper we assume that $\mu$ is finite, that is, $\mu(X)<\infty$ .Next we will present the continuous properties of non-additive measures.Definition 2.2. Let $\mu$ be a non-additive measure on $(X, B)$ .(1) [18] $\mu$ is called null-additive if $\mu(A\cup B)=\mu(A)$ whenever $A$ , $B\in B$ , $A\cap B=\emptyset$ and$\mu(B)=0.$102(2) [19] $\mu$ is called weakly null-additive if $\mu(A\cup B)=\mu(A)$ whenever $A$ , $B\in B,$$A\cap B=\emptyset$ , $\mathrm{u}(A)$ $=0$ and $\mu(B)=0.$(3) [15] $\mu$ is said to be continuous from below (resp. above) if for every increasing (de-creasing) sequence $\{A_{n}\}$ of measurable $\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{s},\mu(\lim_{narrow\infty}A_{n})=\lim_{narrow\infty}4\mathrm{Z}(A_{n})$ holds.We say that a non-additive measure which is continuous from both above and belowis a fuzzy measure.(4) [4] We say that $\mu$ has a pseudo metric generating property (for short ” $\mathrm{p}$ . $\mathrm{g}$ . $\mathrm{p}$ .” ) ifboth $\lim_{narrow\infty}\mu(A_{n})=0$ and $\lim_{narrow\infty}\mu(B_{n})=0$ implies $\lim_{narrow\infty}\mu(A_{n}\cup B_{n})=0$ for$\{A_{n}\}$ , $\{B_{n}\}\subset B.$(5) [5] $\mu$ is said to be exhaustive if $\lim_{narrow\infty}\mu(A_{n})=0$ for any infinite disjoint sequence$\{A_{n}\}\subset B.$(6) [18] $\mu$ is said to be autocontinuous ffom above if for $A\in B$ $\lim_{narrow\infty}\mathrm{u}(A\mathrm{U}73_{n})$ $=\mu(A)$whenever $\lim_{narrow\infty}\mu(B_{n})=0$The class of non-negative measurable functions is denoted by $\mathcal{M}^{+}$ .Definition 2,3. [1, 2] Let $\mu$ be a non-additive measure on (X, B).(1) The Choquet integral of $f\in \mathcal{M}^{+}$ with respect to $\mu$ is defined by$C_{\mu}(f)= \int_{0}$”$\mu_{f}(r)$dr,where $\mu f(r)=\mu(\{x|f(x)\geq r\})$ .$L_{1}^{+}(\mu)$ denotes the class of nonnegative Choquet integrable functions. That is,$L_{1}^{+}(\mu):=$ {$f|f\in$ A $\mathrm{f}"$ , $C_{\mu}(f)<$ $\mathrm{o}\mathrm{p}$ }.103Definition 2.4. [3] Let $f$ , $g\in \mathcal{M}^{+}$ . We say that $f$ and $g$ are comonotonic if $f(x)<7$ $(x’)$implies $g(x)\leq g(x’)$ for $x$ , $x’\in X$ . $f\sim g$ denotes that $f$ and $g$ are comonotonic.The Choquet integral of $f\in M$ with respect to a non-additive measure has the nextbasic properties.Theorem 2.5. [2] Let $f$, $g\in \mathcal{M}^{+}$(1) If $f\leq g$ , then $C_{\mu}(f)\leq C_{\mu}(g)$(2) If $f\sim g,$ then $C_{\mu}(f+g)$ $=C_{\mu}(f)+C_{\mu}(g)$ .3 Regular non additive measuresIn the following we assume that $X$ is a locally compact Hausdorff space, $5\supset$ the class ofopen subsets of $X$ and $C$ the class of compact subsets of $X$ . We suppose that $B$ is theclass of all Borel subsets of $X$ : that is the smallest $\mathrm{a}-$ algebra containing all open subsets,although we may suppose that $B$ is the class of Baire subset, that is, the smallest a-algebracontainning all compact subset, since a non-additive measure has not $\sigma$-additivity. If $X$is metric space, both the class of Baire subset coincides with the class of Borel subset.Definition 3.1. Let $\mu$ be a fuzzy measure on the measurable space $(X, B)$ .(1) $\mu$ is said to be $\sigma$-continuous from below $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 $\mathrm{c}$-continuous from above if $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.(3) $\mu$ is said to be $C$-exhaustive if $\lim_{narrow\infty}\mu(A_{n})$ $=0$ for any infinite disjoint sequence$\{A_{n}\}\subset C.$104First, we define the regular non-additive measures.Definition 3.2. Let $\mu$ be a non-additive measure on measurable space $(X, B)$ . $\mu$ is saidto be inner 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 $i-$ regular if $\mu(C)=\inf$ {$\mu(O)|O\in$ D, $C\subset O$} for all $C\in\not\subset$.$\mu$ is said to be outer regular if $\mu(B)=\inf${$\mu(O)|O\in$ O, $O\supset B$} for all $B\in B.$ Outerregular non-additive measure is called regular if $\mu(O)=\sup${$\mu(C)|C\in$ D, $C\subset O$ } forall $O\in$ D.Proposition 3.3. Let $\mu_{i}$ be an $i$ -regular non-additive measure and $\mu_{\mathit{0}}$ be an O-regularnon-additive measure. $\mu_{i}(O)=\mu_{\mathit{0}}(O)$ for O $\in$ Oif and only if $\mu_{i}(C)=\mu_{\mathit{0}}(C)$ for C $\in C.$Proposition 3.4. Let $\mu_{i}$ be an $i$ -regular non-additive measure and $\mu_{\mathit{0}}$ be an O-regularnon-additive measure. Then we have $\mu_{i}(A)\leq\mu_{\mathit{0}}(A)$ for all A $\in B.$The next two results follow from the definition immediately.Proposition 3.5. Let $\mu$ be a i- (resp. 0-) regular non-additive measure. Then $\mu$ is both$o$-continuous from below, and $c$ -continuous from above.Since i-(0-) regular non-additive measure is $C-$ continuous from above, we have thenext proposition.Proposition 3.6. The i-{p-) regular non-additive measure is c- exhaustive.Next we define another regularity. It is a generalization of completion regularity inclassical measure theory [6]. So we will call it completion regular in non-additive measuretheory. The completion regular fuzzy measure has been studied by [7, 13, 17].Definition 3.7. Let $\mu$ be a non-additive measure on (X, B).105(1) $\mu$ is said to be outer completion regular if for every $\epsilon>0$ there exist an open set$O_{\in}$ such that $\mu(O_{6}\mathrm{S}A)<\epsilon$ .(2) $\mu$ is said to be inner completion regular if for every $\epsilon$ $>0$ there exist a compact set$C_{\epsilon}$ such that $\mu(A\backslash C_{\epsilon})<\epsilon$ .(3) $\mu$ is said to be completion regular if for every $\epsilon>0$ there exist an open set $O_{\epsilon}$ acompact set $C_{\epsilon}$ such that $\mu(O_{\epsilon}\mathrm{z}C_{\epsilon})<\epsilon$ .The next proposition follows from the definition of p.g.p. immediately.Proposition 3.8. Let $\mu$ be a non-additive measure that has a $p.g.p$ . If $\mu$ is both outerand inner completion regular, then $\mu$ is completion regular.Suppose that $\mu$ is a fuzzy measure. If $\mu$ is autocontinous from below, $\mu$ has a p.g.p..Therefore if a fuzzy measure $\mu$ is both outer and inner completion regular and continuousfrom below, then $\mu$ is completion regular [7].Next we will consider the relation among (i-) regularity and completion regularity.The next proposition follows from Proposition 3.6. The proof is essentially similar to[12].Proposition 3.9. Let 72 be a non-additive measure on $(X, B)$ that has $p.g.p$ . If $\mu$ is innerregular, then $\mu$ is completion regular.Proposition 3.10. Let $\mu$ be a non-additive measure on $(X, B)$ that is null-additive andcontinuous from above. If $\mu$ is completion regular, then $\mu$ is regular.The next proposition is a generalization of the result proved in $[8, 14]$ .Proposition 3.11. Let $\mu$ be a fuzzy measure on $(X, B,)$ .(1) If $\mu$ is weak null-additive, then $\mu$ is completion regular.108(2) If $\mu$ is null-additive, $\mu$ is regular.Since null-additivity implies weak null-additivity, a null additive fuzzy measure onlocally compact space is both regular and completion regular.4 Choquet integralThe class of non-negative continuous functions with compact support is denoted by $C_{0}^{+}(X)$ .First we will present the basic properties of Choquet integral with respect to i- (0-)regular non-additive measure.The next proposition follows from monotone convergence theorem of classical measuretheory and definition of Choquet integral.Proposition 4.1. Let $\mu$ be a i-(o-)regular non-additive measure on (X, B). If $f_{n}\downarrow f$ for$\{f_{n}\}\subset C_{0}(X)$ , then $C_{\mu}(f_{n})1$ $C_{\mu}(f)$ .The next proposition is proved by Narukawa et al [9] in the $0$-regular case. The$\mathrm{i}$-regular case is proved similarly.Proposition 4.2. Let $\mu_{1}$ an(l $\mu_{2}$ be i- (0-) regular non-additive measures. If for allf $\in C,(X)C_{\mu_{1}}(f)$ $=C_{\mu_{2}}(f)$ then $\mu_{1}=\mu_{2}$ .Next we discuss the representation of functional on $C_{0}^{+}(X)$ .Definition 4.3. Let I be a real valued functional on $C_{0}^{+}(X)$ . We say that I is comonO-tonically additive iff $f\sim g\Rightarrow I(f+g)$ $=I(f.)+I(g)$ for $f$ , $g\in C_{0}^{+}(X)$ , and that I iscomonotonically monotone iff $f\leq g\Rightarrow I(f)\leq I(g)$ for comonotonic $f$, $g\in C_{0}^{+}(X)$ .If a functional I is comonotonically additive and comonotonically monotone, we saythat I is a $c.a.c.m$. functional.107The next theorem is proved in the similar way to Sugeno et al [16].Theorem 4.4. Let I be a $c.a.\mathrm{c}.m$ . functional on $\mathrm{C}_{0}^{+}(X)$ . We put $\mu_{I}(O)=\sup\{I(f)|f\in$$C_{0}^{+}(X)$ , supp(f)\subset O, $0\leq f\leq 1$ }, $\mu_{I}(C)=\inf\{\mu_{I}(O)|O\in 4\supset, O\supset C\}$ and $\mu_{I}(B)=$$\sup\{\mu_{I}(C)|C\in C, B\supset C\}$ for $O\in$ D,C $\in$ C and $B\in B_{f}$ where supp(f) is a support of$f\in C_{0}^{+}(X)$ .Then $\mu_{I}$ is an $i$ -regular fuzzy measure and $I(f)=C_{\mu I}(f)$ .Since Choquet integral with respect to a non-additive measure is a c.a.c.m functional,applying the theorem above, we have the next proposition.Proposition 4.5. Let $\mu$ be a non-additive measure on $(X, B)$ . There exists an i-regularnon-additive measure $\mu_{r}$ on $(X, B)$ such that $C_{\mu}(f)=C_{\mu},(f)$ for all $f\in C_{0}^{+}(X)$ .Definition 4.6. Let I be a c.a.c.m. functional on $C_{0}^{+}(X)$ . We say that $\mu_{I}$ defined inTheorem 4.4 is an $i$ -regular fuzzy measure induced by $I$ .In the case of z-(0-)regu1ar non-additive measure, the Choquet integral of any measur-able function can be approximated by the Choquet integral of continuous function withcompact support. In the following, we state this fact.The next lemma follows from the definition of regular non-additive measure. Theproof is similar to the case of $\mathit{0}$-regular [11].Lemma 4.7. Let $\mu$ be $a$ i- regular non-additive measure. For every measurable set $A\in B$and an arbitrary $\epsilon>0,$ there exists a continuous function with compact support $f\in$$C_{o}(X)$ and a compact set $C\in C$ such that $C\subset A,$ $1c$ $\leq f$ and $C_{\mu}(f)-\epsilon\leq\mu(C)$ and$|\mu(A)$ $-C_{\mu}(f)<\epsilon$ .Conversely an approximation property in lemma above implies i- regularity.108Lemma 4.8. Let $\mu$ be a non-additive measure on $(X, B)$ . Suppose that $\mu$ has an approx-imation property, that is, For every measurable set $A\in B$ and an arbitrary $\epsilon>0$ ,thereexists a continuous function with compact support $f\in C_{o}^{+}(X)$ and a compact set $C\in C$such that $C\subset A,$ $1c\leq f$ and $C_{\mu}(f)-\epsilon\leq\mu(C)$ and $|\mu(A)$ $-C_{\mu}(f)|<\epsilon$ . Then $\mu$ isi-regular.Applying the lemmas above, we have the next theorem.Theorem 4.9. Let 72 be a non-additive measure on $(X, B)$ , $\mu$ is i- regular if and only iffor every $\epsilon>0$ and $f\in L_{1}^{+}(\mu)$ , there eists a continuous function with compact support$g\in C_{o}^{+}(X)$ and compact set $C\in\not\subset such$ that$C\subset supp(f)$ , $1_{C}\leq 1_{su1\varphi(g)}$ , $|C_{\mu}(g)-\mu(C)|<\epsilon$ and $|C_{\mu}(f)-C_{\mu}(g)|<\epsilon$.Next we will consider additional condition for a c.a.c.m. functional.Definition 4.10. Let I be a c.a.c.m. functional on $C_{0}^{+}(X)$ . We say that I is additivelycontinuous at 0 if and only if $I(f_{n})arrow 0$ and $I(g_{n})arrow 0$ as n $arrow\infty$ for $\{f_{n}\}$ , $\{g_{n}\}\subset C_{o}^{+}(X)$imply $I(f_{n}+g_{n})arrow 0.$We have the next lemma from Definition 4.10 and Theorem 4.4.Lemma 4.11. Let $\mu_{I}$ be a i- regular non-additive measure induced by a c.a.c.m. functionalwith additively continuity at 0. Then $\mu_{I}$ has a $p.g.p.$ .Lemma 4.12. Let $\mu$ be a completion regular non-additive measure on (X, B).(1) For every $A\in B$ and an arbitrary $\epsilon>0$ there exist a continuous function $f\in C_{0}^{+}(X)$such that $C_{\mu}(|1_{A}-f|)$ $<\epsilon$ .(2) If $\mu$ has a $p.g.p.$ , for every simple function $s$ and an arbitrary $\epsilon>0$ there exist $a$continuous function $f\in C_{0}^{+}(X)$ such that $C_{\mu}(|s-f|)<\epsilon$ .I QEILet $f$ be a bounded non-negative measurable function. There exists a sequence $\{s_{n}\}$of simple function such that $s_{n}\uparrow f$ as $narrow\infty$ . Applying monotone convergence theoremin classical measure theory to non decreasing function $\mu(|s_{n}-f|>\alpha)$ , we have the nextproposition.Proposition 4.13. Let $\mu$ be a completion regular non-additive measure on $(X, B)$ . If $\mu$has a $p.g.p.$ , for every measurable function $f\in L_{1}^{+}(\mu)$ and an arbitrary $\epsilon>0$ there exista continuous function $g\mathrm{E}$ $C_{0}^{+}(X)$ such that $C_{\mu}(|f-g|)<\epsilon$ .Let I be a c.a.c.m. functional on $C_{0}^{+}(X)$ with additively continuity at $0$ . Since theinduced regular non-additive measure $\mu I$ has a p.g.p. and $C-$ exhaustivity, it follows fromProposition 3.9 that $\mu_{I}$ is completion regular. Therefore we have the next theorem.Theorem 4.14. Let I be a $c.a.c.m$. functional on $C_{o}^{+}(X)$ with additively continuity at 0and $\mu_{I}$ be $a$ induced $i-$ regular non-additive measure. Then for every measurable function$f\in L_{1}^{+}(\mu)$ and an arbitrary $\epsilon>0,$(1) there exists a continuous function $g$ with compact support such that$|C_{\mu I}$ $(f)$ $)-C_{\mu_{I}}(g)|<\epsilon$,,(2) there eists a continuous function $g$ with compact support such that$C_{\mu I}(|f-g|)<\epsilon$ .References[1] G.Choquet Theory of capacities Ann. Inst. Fourier, Grenoble. 5 (1955) 131-295.[2] D. Denneberg, Non Additive Measure and Integral, Dordorecht:Kluwer AcademicPublishers, 1994.10[3] C. Dellacherie, Quelques commentaires sur les prolongements de capacit\’es, S\’eminairede Probabilit\’es $\mathit{1}\mathit{9}\theta \mathit{9}/\mathit{1}\mathit{9}$ 70, Strasbourg, Lecture Notes in Mathematics, 191(Springer, 1971) 77- 81.[4] I. Dobrakov, On submeasure $\mathrm{I}$ , Dissertationes mathematicae, 112, 5-35, (1974).[5] L. Drewnowski, Topological rings of sets, continuous set functions, integration $\mathrm{I}\mathrm{I}$ .Bull. Acad. Polon. Sci Ser. Math. Astronom. Phys. 20. 277-286, (1972)[6] P. R. Halmos, Measure theory, New York: Van Nostrand, (1950).[7] A. B. Ji, Fuzzy measure on locally compact space, The Journal of Fuzzy Mathematics,Vol. 5, No 4.,(1997) 989-995.[8] J. Li, M. Yasuda, Lusin’s theorem on fuzzy measure spaces, Fuzzy sets and systemsto appear.[9] Y. Narukawa, T. Murofushi, M. Sugeno, Regular fuzzy measure and representationof comonotonically additive functionals, Fuzzy Sets and Systems 112,(2),(2000),177-186,[10] Y. Narukawa, T. Murofushi, M. Sugeno, Boundedness and Symmetry of ComonO-tonically Additive Functionals, Fuzzy Sets and Systems 118, (3), (2001), 539-545.[11] Y. Narukawa, T. Murofushi, $\mathrm{R}\mathrm{e}\mathrm{g}\mathrm{u}_{\mathrm{Q}}1\mathrm{a}\mathrm{r}$ Non-additive measure and Choquet integral,Fuzzy Sets and Systems, to appear.[12] E. Pap, Regular null additive monotone set functions, Univ. $u$ Novom Sadu $Zb$ . rad.Prorod.-Mat $Fak$. Ser. mat. 25, 2 $(\mathit{1}\mathit{9}\mathit{9}\mathit{5})_{f}\mathit{9}\mathit{3}- \mathit{1}\mathit{0}\mathit{1}$[13] E. Pap, Null-Additive Set Functions, Dordorechet: Kluwer Academic Publishers,1995.111[14] J. Song, J. Li, Regularity of null additive fuzzy measure on metric spaces. Interna-tional Journal of General Systems, 32 (3) (2003) 271-279.[15] M. Sugeno, Theory of fuzzy integrals and its applications, Doctoral Thesis, TokyoInstitute of Technology, (1974).[16] M. Sugeno, Y. Narukawa and T. Murofushi, Choquet integral and fuzzy measureson locally compact space, Fuzzy Sets and Systems, 99 (2) (1998) 205-211.[17] J. Wu, C. Wu, Fuzzy regular measures on topological spaces. ffizzy sets and Systems,119 ,(2001) 529-533.,[18] Z. Wang, The autocontinuity of set function and the fuzzy integral. J. Math. Anal.Appl., 99, (1984), 195-218.[19] Z. Wang, G. J. Klir, Fuzzy Measure Theory, New York: Plenum, 1992.

正則非可法的測度について　(情報科学と函数解析の接点 : これまでとこれから)

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正則非可法的測度について　(情報科学と函数解析の接点 : これまでとこれから)

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正則非可法的測度について (情報科学と函数解析の接点 : これまでとこれから)

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正則非可法的測度について　(情報科学と函数解析の接点 : これまでとこれから)