The Australian Journal of Mathematical Analysis and Applications

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1 The Australian Journal of Mathematical Analysis and Applications Volume 6, Issue 2, Article 9, pp. 1-8, 2009 ON THE PRODUCT OF M-MEASURES IN l-groups A. BOCCUTO, B. RIEČAN, AND A. R. SAMBUCINI Received ; accepted 18 July, 2008; published December, DIPARTIMENTO DI MATEMATICA E INFORMATICA, VIA VANVITELLI, 1 I PERUGIA ITALY boccuto@dipmat.unipg.it URL: KATEDRA MATEMATIKY, FAKULTA PRÍRODNÝCH VIED, UNIVERZITA MATEJA BELA, TAJOVSKÉHO, 40 SK BANSKÁ BYSTRICA SLOVAKIA riecan@fpv.umb.sk DIPARTIMENTO DI MATEMATICA E INFORMATICA, VIA VANVITELLI, 1 I PERUGIA ITALY matears1@unipg.it URL: ABSTRACT. Some extension-type theorems and compactness properties for the product of l- group-valued M-measures are proved. Key words and phrases: l-group, extension, weak σ-distributivity, product measure, countable compactness Mathematics Subject Classification. 28B15. ISSN electronic: c 2009 Austral Internet Publishing. All rights reserved. This paper was supported by project SAS/CNR Slovak Academy of Sciences/ Italian National Council of Researches "Integration in abstract structures" 2007/09, grant VEGA 1/0539/08, grant APVV LPP and GNAMPA of CNR..

2 2 A. BOCCUTO AND B. RIEČAN AND A. R. SAMBUCINI 1. INTRODUCTION In the study of probability theory, in many applications it is advisable to deal with set functions, which are not necessarily additive, but satisfy other properties: for example, continuity from below and from above for sequences of sets and "compatibility" with respect to the operations of finite suprema and infima. These functions are called M-measures see [7, 12, 21]. For example, in decision making, this is the case of the theory of intuitionistic fuzzy events shortly IF-events, which are pairs A = µ A, ν A of measurable functions µ A, ν A : Ω [0, 1] such that µ A +ν A 1. For a literature about IF-sets, see [1, 2, 5, 6, 12, 13, 17, 18, 19]. Another application is the theory of joint random variables: in this context the M-measure extension theorem plays a crucial role in the construction of joint observables. Moreover, to consider latticegroup or Riesz space-valued set functions allows to get applications in stochastic processes and in probabilities depending on the time and/or on the informations of the individual. In this paper we continue the investigation dealt with in [4] and, starting from an extensiontype existence theorem for M-measures with values in l-groups, we obtain existence results in the countably compact case for M-measures and product of M-measures. We begin with the following 2. PRELIMINARIES AND BASIC RESULTS Definition 2.1. An l-group lattice ordered group R is said to be 2.1.1: Dedekind complete if every nonempty subset of R, bounded from above, has supremum in R; 2.1.2: super Dedekind complete, if it is Dedekind complete and for any nonempty set A R, bounded from above, there exists a countable subset A A, having the same supremum as A : A bounded double sequence a i,j i,j in R is called regulator or D-sequence if, for each i N, a i,j 0, that is a i,j a i,j+1 for all j N and a i,j = : Given a sequence r n n in R, we say that r n n D-converges to an element r R if there is a regulator a i,j i,j, such that to every map ϕ N N there corresponds a positive integer k with r n r a i,ϕi for all n k. In this case, we write D lim n r n = r or simply lim n r n = r, since no confusion can arise. Definition 2.2. We say that R is weakly σ-distributive if, for every D-sequence a i,j i,j, a i,ϕi = 0. ϕ N N Remark 2.3. Observe that in weakly σ-distributive l-groups D-convergence for sequences coincides with order convergence. An example of a super Dedekind complete weakly σ- distributive l-group is the space L 0 Y, Σ, ν, where Y, Σ, ν is a measure space with ν σ- additive and σ-finite, see [14]. From now on, let X be a set and W be an algebra of subsets of X. j N

3 ON THE PRODUCT OF M -MEASURES IN l-groups 3 Definition 2.4. A family A of subsets of X is called monotone class if the following properties hold: a: n N A n A for every non-decreasing sequence A n n N in A, b: n N A n A for every non-increasing sequence A n n N in A. Definition 2.5. A family K of subsets of X is called countably compact class, if for every sequence C n n of elements of K we get n C i whenever C i for any n N. i N Definition 2.6. A set function λ : W R is said to be countably compact, if there is a countably compact class K with the property that for any A W there corresponds a D- sequence a i,j i,j such that to any ϕ N N two sets B W, C K can be found, with B C A and λa \ B a i,ϕi. In the sequel we will use the following fundamental results [8, 9, 10, 20]. Lemma 2.7. [20], Theorem Let {a n i,j i,j : n N} be any countable family of regulators. Then for each fixed element u R, u 0, there exists a regulator a i,j i,j such that, for every ϕ N N, u a n i,ϕi+n a i,ϕi. Theorem 2.8. [11], Theorem 1.6.B If A PX is a monotone class of sets such that c: W A, then A includes the σ-algebra of subsets of X σw generated by W. Lemma 2.9. [10], Lemma 413R Let K be a countably compact class of sets. Then there is a countably compact class K K such that K L K and n N K n K whenever K, L K and K n n is a sequence in K. Definition A set function µ : W R is called M-measure if it satisfies the following properties: 2.10.i: µ = 0; 2.10.ii: µa B = µa µb = supµa, µb for all A, B W; 2.10.iii: µa B = µa µb = infµa, µb for any A, B W; 2.10.iv: µ is continuous both from below and from above, that is: if A n A, resp. B n B, A n, A B n, B W, n N, then µa n µa µb n µb. It is known that Theorem [4], Theorem 3.1 Let R be a super Dedekind complete weakly σ-distributive l-group. For every bounded R-valued M-measure µ, defined on a ring W, there is a unique M-measure µ defined on the σ-ring σw generated by W, extending µ. The line of the proof of this theorem is the following: at the first step, the M-measure µ is extended to W +, which is the class of all sets A of the type A := A n, with A n A n+1, A n W for all n N,

4 4 A. BOCCUTO AND B. RIEČAN AND A. R. SAMBUCINI by setting µ + A = lim n µa n the limit exists in R and does not depend on the choice of the sequence A n n. Successively, it is extended to W, which is the ideal generated by W +, by setting µ A = inf{µ + B : B W +, B A} for every A W. Then it is proved that µ := µ σw is an M-measure, extending µ. Finally, in order to prove uniqueness, observe that, if ν : σw R is an M-measure with extends µ, then it coincides with µ on a monotone family containing W, and this concludes the assertion. We denote by µ : σw R this extension. Observe that: Theorem If µ is countably compact, then µ is countably compact too. Proof: Let K be the countably compact class related with countable compactness of µ and K K be a countably compact class associated with K, closed with respect to finite unions and countable intersections, existing by virtue of Lemma 2.9. Set 2.1 L = {A σw : there is a regulator a A i,j i,j such that ϕ N N B A σw, C A K : B A C A A and µa \ B A a A i,ϕi }. It is enough to prove that L satisfies Theorem 2.8. First of all, the inclusion W L follows directly from the definition of countably compact measure. We now prove that L is a monotone class. If A n n is a non-decreasing sequence in L, let A = n N A n. Since µ is continuous from below, there is a regulator b i,j i,j with the property that to every ϕ N N there corresponds a positive integer k with µa \ A n b i,ϕi whenever n k. By 2.1, in correspondence with A k let a k i,j i,j be the associated regulator, B k σw, C k K be such that B k C k A k A and Set c k i,j = 2ak µa k \ B k a k i,ϕi. i,j + b i,j, i, j N. The double sequence c i,j i,j is a regulator. We obtain: µa \ B k µa \ A k + µa k \ B k c k i,ϕi ; moreover, by Lemma 2.7, there exists a D-sequence f i,j i,j such that µa \ B k µω f i,ϕi. c n i,ϕi+n Let now A n n be a non-increasing sequence in L, and set A = n N A n. In correspondence with A n, let B n, C n, a n i,j i,j be as in 2.1. Set B = n N B n, C = n N C n. Then there is a positive integer p such that µa \ B µa p \ B p a n i,ϕi+n a p i,ϕi+p and, by Lemma 2.7, there exists a D-sequence g i,j i,j such that µa p \ B p µω g i,ϕi.

5 ON THE PRODUCT OF M -MEASURES IN l-groups 5 Moreover B C A, B σw and C K, since K is closed with respect to countable intersections. This concludes the proof. 3. EXISTENCE OF PRODUCT MEASURES Let R be a super Dedekind complete weakly σ-distributive l-group, X, S, µ, Y, T, ν be two measure spaces, where S, T are algebras and µ, ν are R-valued countably compact M- measures. We want to define the product measure of µ and ν. By Theorem 2.12 there exist two countably compact M-measures µ : σs R, ν : σt R, extending µ and ν respectively. Let now E be the family of the elementary sets, of the type n 3.1 A l B l, where n N, A l σs, B l σt, l = 1,..., n, and A l B l A s B s = whenever l s. We prove the following: Theorem 3.1. Let µ, ν be R-valued countably compact M-measures as above. Then there is exactly one countably compact M-measure κ : σe R with 3.2 κa B = µa νb for all A σs, B σt. Proof: Set κa B = µa νb, A σs, B σt. Let now E = A B, F = C D σs σt. We get: κe F = κ A B C D = κ[a \ C B A C B D C \ A D] = µa \ C νb µa C νb D µc \ A νd = µa \ C νb µa C νb µa C νd µc \ A νd = µa νb µc νd = κe κf. Analogously, it is possible to check that κe F = κe κf for all E, F σs σt. If E, F E, the analogous results follow by virtue of the distributive laws. So, κ : E R can be defined as follows: n n κ A l B l = µa l νb l. Now, in order to prove that κ is countably compact, we show that for each A E a D- sequence c i,j i,j can be found, with the property that: 3.3 ϕ N N, B E, C K with B C A and κa \ B c i,ϕi. As µ and ν are countably compact measures, there are countably compact classes K 1 PX, K 2 PY, with the following property: for all A σs and B σt there is a regulator

6 6 A. BOCCUTO AND B. RIEČAN AND A. R. SAMBUCINI α i,j i,j such that ϕ N N E σs, C K 1, F σt, D K 2, with E C A, F D B, sup{µa \ E, νb \ F } α i,ϕi. Set now H = {E = C D : C K 1, D K 2 }. It is easy to see that H is countably compact. By [10], Lemma 451H and Lemma 2.9 there exists a countably compact class K containing H, and closed with respect to finite unions and countable intersections. Let A be any element of E. There exist n N, A l S, B l T, l = 1,..., n such that A = n A l B l. Since µ, ν are countably compact M-measures, then to l = 1,..., n there correspond two D-sequences a l i,j i,j, b l i,j i,j and H l K 1, G l K 2, E l, F l S with E l H l A l, F l G l B l, µa l \ E l νb l \ F l a l i,ϕi+l, b l i,ϕi+l. By Lemma 2.7, there are two regulators a i,j i,j, b i,j i,j, with [µx] a i,ϕi, [νy ] b l i,ϕi+l a l i,ϕi+l b i,ϕi for all ϕ N N. Put c i,j = a i,j b i,j, i, j N, and n C = H l G l, B = n E l F l : we get B C A, C K, and [ n ] [ n ] κa \ B = κ A l B l \ E s F s s=1 [ n ] [ n ] = κ A l \ E l B l A s B s \ F s = s=1 [ n ] [ n ] µa l \ E l νb l µa s νb s \ F s s=1 s=1 [ n ] [ n ] µa l \ E l νb s \ F s c i,ϕi. Countable compactness of κ follows. We now prove 2.10.iv, namely that κa n 0 resp. κb n κb whenever A n n is a non-increasing sequence in E, with A n = resp. B n E, n N, B n B, B E. n N

7 ON THE PRODUCT OF M -MEASURES IN l-groups 7 Pick now arbitrarily any sequence A n n in E, with A n. Since κ is countably compact, then in correspondence with each positive integer n a D-sequence d n i,j i,j and two elements B n E, C n K can be found, with B n C n A n and κa n \ B n d n i,ϕi+n. Again by Lemma 2.7, there exists a D-sequence d i,j i,j with the property that [κx Y ] d i,ϕi. Set now D n = n C l. We get d n i,ϕi+n A n =. D n n N n N Since K is a countably compact class, a positive integer m can be found, with m m B l D m = C l =. For each n m we obtain [ m ] m κa n κa m = κ A m \ B l = κ A m \ B l m m κ A l \ B l = κa l \ B l d i,ϕi and then lim n κa n = 0. Let now B n E n N, B n B, B E. Then B \ B n, and hence Thus we get: κb = κb \ B n B n = κb \ B n κb n κb \ B n κb i. κb lim n κb \ B n κb i = κb i κb, and hence κb = lim i κb i. Furthermore, if C n C, then C n \ C, and κc n = κc n \ C C = κc n \ C κc, κc n = κc n \ C κc = 0 κc = κc. Thus we proved that κ is an R-valued countably compact M-measure, defined on E. By Theorem 2.12 there is a unique countably compact M-measure κ, defined on σe and extending κ.

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