Choquet积分的收敛定理

第 15卷第 2期 200】年 6月 模 糊 系 统 与 数 学 Fuzzy Systems and M athematics Vo1．15，No．2 Jun．，2001 Article ID：1001—7402(200])02-0051一g4 Convergence Theorems of the Choquet Integral GUo Cai—mei ．ZHANG De-IF (1·Department of Basic Science，Changchun University，Changchun 130022 ．China 2 Depa rtment of Computer Science．Jilin University，Changchun 130023㈣Chi ) Abstract：So fa r·the Choquet integral with respect to a fuzzy ~leasu re ha h n tudi d uch e ten vP by Dr M urofu i’Sugeno and F,ome othe rs ． But the convergence theory is not enough The pape r|㈣ i is Ju t d scuss1ng this topic·and~hows∞ 0nvergence theorems for the Choquet integra1．These in Lude generalized R?Gnotone convergence theorem·Fatou。 lerrl[I]8s，etc We conclude that the Choquet inte ra1 has the same convergence theorenls as Sugeno integral o'wns ． Key words：M easu re Theory；('hcxiuet Integral；Fuzzy Measure：Fuzzy Integral}Convergence The[1re】lL CLC number：(I1 74．1 2 Document code．A 1 Introduction Since Sugeno introduced the concept of f uzzy measures in 1 974．the fuzzy integral with respect 1o a fuzzy measure is generally developed． such as Sugeno ．Ralescu ．Zhao ”一．Suarez!。． W u ．etc．The common property of all the vari OOS kinds of fuzzy integrals is that they are monc~ tonic functional from the space of nonnegat[ve measurable functions to E0，。。] Just based on the property，M urofushi and Sugeno also view the Choquet integral which x~-as introduced by Choquet[ ]in 1 953．as a kind of fuzzy,integrals． and have given a much extensive study in E3 6]． It ls well—known that convergence theorems a very important in classical integral theory． But to the Choquet integra1．convergence theory is not enough．Since so far，only the monotone convergence theorem ls shown ． W hethcr can other convergence theorems be established? The answer is JLJS1 the paper s purpose． In the paper．we will show various kinds of gcnerali~d convergence theorems of the Choquet integra1．these include gegeralized monotone COIl— vergenee theorem ，generalized Fatou s 1emmas． etc． The rest of the paper is divided into two parts． Section 2 will give some concepts and reviews on fuzzy measures and the Choquet inte gra1 as preparation．Section 3 wil1 show the main results of this pape r． 2 Fuzzy measures and the Choquet integral I et be a nonempty classical set． a o-algebra formed by the subsets of ．( ， ) the measurable space． Definition 2．1 Let ： 一E0．。。]be a set f1】nct】on．Then ：麓 口n da do nin OSZ ；KE I．⋯of SE e of J ilinU uive r sgy ⋯ 。 noil—additireintegraIs，ZHANG DeIIt1 964 )．male．born n Nong’an JI1 p~ofessor and vice—president of Jgin Prov inst。t Educatlon． f rcn1[恒 on ：ugzy analvsts 维普资讯 http://www.cqvip.com 模 糊 系 统 与 数 学 200]正 (1)be is said to be empty null if be(0)一_0； follow (2)beis said to be monotone ifbe(A)≤be(B) whenever A[ B； (3)be is said to be continuous form below if ( )十 ( )whenever A 十A； (4) is said to be conditionally continuous form above_f (A ) (A)whenever ，．+A． and be(A )<。e for a fixed n-≥ l_ Definition 2．2一 A set function be：一e／一 [0，。。]is called a fuzzy'Ineasure if it is empty null and monotone． The triplet f。X ． ≯ ．日j is called a fuzzy measure space．be is said to be continuous．if it is both continuous from below and conditionally continuous from above． I eI the symbol M ‘ )denote the set of all fuzzy measures on (X一。 )． Proposition 2．1～ Let{ ．．}be a sequence of fuzzv measures on(X，。 )1．c．{ } M tX)． and a set function from √ to[0．。。：．If{ ．．} converges to be(setwise convergence，i．e． pr,．(．1j 一 t^ )for every ^∈ 。，simply wrhc as — J．then be is a fuzzy measure． Remark 2．1 In this proposition． if , gA．． (，?≥1)is continuous．and 一 ．then it doesn 【 imply that is continuous． (see a counter example given by Zhang ．) Proposition 2．2 Let{ ．．}be a．sequence of fuzzy measures on (X． 。，)．and be a set— function from。 to[0．∞]．If{ ．．}converges uniformly to ，then ( ≥ 1)is continuous implies th “is conlinuous． A function f：X一 [0．。c]is said to be ．measurable if f ( )∈ 。，for each B∈ Borel([0．。。])． All the measurable function from 二， to Ln．∞]is denmed hy F(x) Definition2 3t I et／ ∈ F(JY)． ∈ M ( )． Then the Choque1 integral of／ with respect to ．which denoted by‘c)Ifdbe is as (c)I —l be(f>t)dt W here the right side integral is Lebesgue integra1．and(厂>r)stand for{ ∈X：，( )>f}． The propositions of the Choquet integral refered to r1．4]． Definition 2．4一 A set N ∈ ，is calted a null—set with respect to iff (AUⅣ)一_ ( )· for all A∈ ． By using the null set ．the 'almost ever— where concept is defined as! J尸‘ )a．e Ine&ns that there【s a nuIl set N ．such that P(』) t rue for all ∈ ．where P 【 ) is a proposition concerning the points of X． 3 Convergence theorems rheorem 3．1 I et{ ．}[F(X)．／∈I-"t j (1) Let be be cont Jntlous from below If 』，．‘』a．e．．then (f){ dbe十( )Ifdbe t2)l et he conditionally continuous from above．If ． ／a．e．，and iff ≤F a．e．for some r g∈F(x)with(c)Igdbe< ∞，then (C) ． [fod (C)J fd~ Lemma 3．1 I e1 I [ M (X)． ∈ M (X)， ／-∈ F(X)． (1)If ．．十 ，then (c)lfd 十(c)Ifdbe (2)If ，and (c)ll厂d < 。。for sortie ”≥ 1 lhen (c)Jfd／J．． (C)Jfd~ Theorem 3．2 I et{ ．)[ F(X)．f∈ F(x)， [ M (X)． ∈M (X)．If A 十 ． 十 ．then is continuous from below implies ‘cj 1 ．d ．十(c)Ifdp, 维普资讯 http://www.cqvip.com 第 2期 GUO Cai—mei，ZHANG De一 Co rgeⅢ Theorems of rhe Choquer Int gH】 53 Theorem 3．3 Let{A }CF(X)，，∈F(X)． r 1 cM( )， ∈M( )，and(c)l d <。。 for some ≥l_If ．v f a．e．， + ，then is conditionally continuous form above implies tf1at r (c)l ．d +(c){／d J J Theorem 3．4 I et i )[ F(X)， ．． [ M tx 1．If 堕 is continuous from below．then r (c)f(1imf．】d(Iim )≤liIn(c)l／ d ～ ll — l y J Theorem 3．5 I．et( ．}二F‘X J． }[ M(X)．and(c)l(sup )d(sup )<。。for s。f【1c J ^ ≥ 1．If lim is conditionally continuous from above，then 【 ) d，‘≤ t[1_)i(1ira．(．)d‘Tin1 ‘j 。 J ● · Theorem 3 6 、．el tj ，=．FtX 3 1∈ Fl )．{ ．．}二 (X)．,ll∈ ( )，and is bo!h continuous from below and conditionally contintl F( )． ∈M( )．If(c)llfo～fld — 0("— ，·then we say that{， is C—mean convergent to f．It is simply written by fo— _，(f m)． Definition 3．2。 1 Let j ＼CFtx) {∈ F( )， ∈M ( )．Iflim,a(。 一_厂l≥ e)一0for ever),￡> 0，then we say{j ＼converges to f in fuzzy measure ． In short．it is denoted by 三 { Theorem 3．7 Let i [ F(X)．f∈ Ftx)．If fo一 }c m)，then j ． l'heorem 3 8 I．ct i j[ F tX )， ∈ Ff )． ∈ M ( j． ( )= < 。。．If there exists QF(X)．(c)f算d <∞，such that lJ ～ff≤ g for each 1．then j j implies j 一 L m Proof By the definitlon of the Choquet integra1．for each ≥¨ 1．we have “’) _，1．一厂d —j (Jfo—fl>，Jd， r (】us from above．If Lc)f(supfDd(sup,％)< but ⋯ 一 一 ⋯ for some ≥1．th ／．．一厂． 一 impllcs I (Jfo～fl> 1)d (f)I／ d 一(c)Ijd[x Corollary 3．1 Let{ }[ F(X )， ，}[ M (X)． (1)If is continuous from below．then (c)f(1imf,)d ≤lim(C) ．d (2) If is conditionally continuous from above，{(sup^ )dp< ∞ for some ≥ 1．then Iim(c)I d ≤(c)l(1iraf,．)d Coroll~try 3．2 IA f∥[F(X)．{ [ M 【 )．Then 【1)((、)I／d(1lm )≤ lim(c)Ifd (2)If((、，1 (sup,u j<∞ for some ≥1， th n ¨f 一，≥ ([fo—f『≥ )出+f” (1 一厂≥ejd By making use of{ } e lim (1 一 ，『≥ ￡j= 0 for V e> 0 and l 一／I≤g for each ≥ 1，from Lebesgue dominated convergence theorem of Lebesgue inte： gra]．vee have Ⅳ lirai ‘『 一厂 ≥ E)df=0 n o=J By (1 ．一f『≥f)≤ ( )=M 题 快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题 ．给出 Choquet积分的一些收敛定理 ，包括广义单 调收敛定理、Fatou引理等。从 中可以看出，Choquet积分与 Sugeno模糊积分具有相同的收敛定理 关键词：和4度论；Choquet积分；模糊测度；模糊积分；收敛定理 维普资讯 http://www.cqvip.com