A decomposition theorem for maxitive measures

A DECOMPOSITION THEOREM FOR MAXITIVE MEASURES P A UL P O N C E T A B S T R AC T . A maxitive measure is the analogu e of a finitely additiv e mea sure or charge, in which the u sual add ition is replaced by the sup remum oper ation. Con- trarily to charges, maxitive measures often h a ve a density . W e show that m axiti ve measures can be deco mposed as the sup remum of a maxitive me asure with density , and a residual maxiti ve measure tha t is null o n compact sets un der specific con ditions. 1. I N T R O D U C T I O N In the area of idempotent analysis, maxitiv e measures are usually known as idempo- tent measur es after Maslov [23]. Maxitive measures are defined analogously to finit ely additive measures with the supremum operation ` in place of the addition  . In the literature, they first appeared in an article by Shi lkret [33], and then hav e been re- discovered and explored for the purpos e of capacity theory and large deviations (e.g. Norber g [26 ], O’Brien and V erv aat [27], Gerritse [13], Puhalski i [32]), idemp otent analysis and max-plus (tropical) algebra (e.g. Maslov [23], Bellalouna [8], Akian et al. [4], Del M oral and Do isy [11], Akian [3]), fuzzy set theory (e.g. Zadeh [36], Sugeno and Murofus hi [34], Pap [28], De Cooman [9], Nguyen et al. [24], Poncet [30]), o pti- misation (e.g. Barron et al. [6], Acerbi et al. [1]), or fractal geometry (Fa lconer [12]). Let E be a n onempty set. A pr epaving on E is a collection of subsets of E con- taining the empty set and clo sed un der finite unions. As sume i n all the sequel that E is a prepa ving on E and that L is a partially ordered set or poset wih a bottom ele- ment, that we denote by 0 . An L -valued maxitive measur e (resp. completely maxitive measur e ) o n E is a map ν : E Ñ L such that ν pHq  0 and, for ev ery finite (resp. arbitrary) family t G j u j P J of elements of E such th at  j P J G j P E , the supremum of t ν p G j q : j P J u exists a nd satisfies ν p ¤ j P J G j q  à j P J ν p G j q . If we take for E th e prepaving of all finite subsets o f E , then e very maxitiv e measure ν on E can be written as (1) ν p G q  à x P G c  p x q , Date : Decembre 28, 2009 . 1991 Mathematics Subject Classification. Prima ry 28B15, 28C15; Secondary 06B35, 03E72, 49J52. K ey wor ds an d ph r a ses. max -plus alg ebra, max iti ve measures, idempotent measu res, ca pacities, con- tinuous posets, domains, continuo us lattices. 1 where c  p x q  ν pt x uq , s ince G   x P G t x u , where the uni on runs over a finite set. W e say t hat c  is a car dinal density (or a density for s hort) of ν w hen Equati on (1) is satisfied. With this s imple example, where E need not to be finit e for ν t o have a density , we see why compelling E P E w ould be inappropriate. In the general case, s ingletons t x u do not necessarily belong to E , but, as we shall see, one t o extend maxitive measures to the whole power set 2 E under mild conditions , so it is tempt ing to consider c  p x q :  ν  pt x uq instead, where ν  is th e extension of ν defined as in Equ ation (3) below . This i dea, which appeared in [17, 18] and [3], will indeed lead us to n ec essary and suffi cient conditions for a maxit i ve measure to have a density (see Theorem 3.1). In this article, we are interested i n decom posing a maxitive measure into a r e gular part, which is a m axiti ve measure wi th a cardinal dens ity , and a r esid ual part, also maxitive, and null on compact sets under specific con ditions. Our m oti v ation comes from possibl e applications to Radon-Nikodým like theorems for the S hilkr et inte gral , dev el oped in [33] for maxiti ve m ea sures, also kno wn as Maslov’ s idempotent inte gral . The results we shall g i ve on maxitive measures are stated i n the general case where these measures take their va lues in a domain , t he d efi nition of w hich follows. (For more background on domain theory , see the monograph by Gier z e t al. [15].) A subset F of a poset p P , ¤q is filter ed if, f or all x, y P F , one can find z P F such that z ¤ x and z ¤ y . A filter of P i s a n onempty filtered s ubset F of P such that F  t y P P : D x P F , x ¤ y u . W e s ay that y P P is way-above x P P , writt en y " x , if, for ev ery filter F with an i nfimum  F , x ¥  F implies y P F . The way-above r elat ion , useful for studying lattice-valued upper semicontinuo us functi ons (see Gerritse [14] and Jonasson [19]), is dual to the usu al way-below re lation , b ut is more appropriate in our cont e xt. Coherently , our notio ns of continuous posets and domains are dual to the traditi onal ones. W e thus say that th e pos et P is continuo us if Ò Ò x :  t y P P : y " x u is a filter and x   Ò Ò x , for all x P P . A domain is a continuous poset in which e very filter has an infimum. A poset P has the interpolati on pr operty if, for all x, y P P , if y " x , there exists some z P P such th at y " z " x . In continuous p osets it is well known t hat the in terpolation property hol ds, see e.g. [15, Theorem I-1.9]. This is a crucial feature that is behin d m an y i mportant results of the theory . W ell known examples of domains are R  , R  , and r 0 , 1 s . For t hese posets, t he way-abov e relation coincides with the strict order ¡ (except perhaps at the top). These posets are common ly used as t ar get sets for maxit i ve m ea sures, and many trials were made for replacing them by more general ordered structures (see G re co [16], Liu and Zhang [22], De Coom an et al. [10], Kramosil [20]). Nevertheless, the importance of the continuity assumpt ion of these structures for applications to id empotent analys is or fuzzy set theory has been identified lately . Pioneers i n thi s direction were Aki an (see [2], [3]) and Heckmann and Huth [17, 18]. See Lawson [21 ] for a survey on the use of domain theory in idempotent mathematics. See also Poncet [31] and references therein. 2 The paper is organized as follows. Se ctions 2 and 3 im pro ve results of [3] and [17, 18]: we give a representation theorem for maxitiv e measures, deri ve t he extension theorem cited abov e, and revisit the problem of finding necessary and suf ficient condi- tions for a maxitive measure to ha ve a cardinal de nsity . Section 4 is ne w and states the announced decomposition theorem. 2. R E P R E S E N T I N G M A X I T I V E M E A S U R E S B Y I D E A L S An ideal of t he prepaving E is a nonempty subset I of E which is st able under finite unions and such that, if A  B and B P I , then A P I . The next proposition , inspi red b y Nguyen et al. [25], provides a generic way of constructing a maxitive measure from a nondecreasing f amily of ideals. Pr oposition 2 .1. Let p I t q t P L be some family of i deals of E such that , for all G P E , t t P L : G P I t u is a filter with infimum. Define ν : E Ñ L by (2) ν p G q  © t t P L : G P I t u . If p I t q t P L is right-continuo us, in th e sense that I t   s " t I s for al l t P L , then ν is maxitive. Remark 2.2. Ass uming that t t P L : G P I t u is a filter for all G P E makes the family p I t q t P L necessarily nondecreasing. Pr oof. L et ν be give n by Equation (2). Obviously , ν is order- preserving, so it remains to show that, for a ll finite f amily t G j u j P J of elements of E , and for e very upper bound m P L of t ν p G j qu j P J , we get m ¥ ν p  j P J G j q . Let s " m . One has G j P I s for all j P J , thus  j P J G j P I s . Thi s implies  j P J G j P  s " m I s  I m . Eventually m ¥  t r P L :  j P J G j P I r u  ν p  j P J G j q , so ν is maxit i ve.  Supposing the continuity of the range L of the maxitive measure enables us to re- move the assumption of right-continuity of the fa mily of ideals and gi ves the con verse statement as follows. Pr oposition 2.3. Assume that L is a continuous poset. A map ν : E Ñ L is a maxitive measur e if and o nly if ther e is some family p I t q t P L of ideal s of E such that, for all G P E , t t P L : G P I t u is a filter with infimum and ν p G q  © t t P L : G P I t u . In this case, p I t q is right -c ontinuous if and o nly if I t  t G P E : t ¥ ν p G qu for all t P L . Pr oof. If ν i s maxitive , simpl y take I t  t G P E : t ¥ ν p G qu , t P L , which is right-continuous sin ce L is continuous. Con versely , assume that Equation (2) is satisfied. Let J t   s " t I s . p J t q t P L is a n ondecrea sing family of ideals of E such that J t  I t for all t P L . Moreov er , p J t q t P L is right-continuous thanks t o the interpolation property , and by continuity of L one has ν p G q   t t P L : G P J t u . Using Proposition 2.1, ν is maxitive. 3 Assume that p I t q is right-conti nuous. T he inclu sion I t  t G P E : t ¥ ν p G qu is clear . If t ¥ ν p G q , we want to sho w that G P I t , i.e. G P I s for all s " t . So let s " t ¥ ν p G q . Equati on (2) im plies that G P I s , and the inclusion I t  t G P E : t ¥ ν p G qu is prove d.  From Proposition 2.3 we can deduce the foll o wing corollary , which m ost of the ti me enables one to extend a maxitive measure to the entire power set 2 E . This is a slight i m- provement of Heckmann and Huth [18, P roposition 12] and Akian [3, Proposition 3.1], the latter being inspired by Maslov [23, Theorem VII I-4.1]. Henceforth, E  denotes th e collection of all A  E such that t G P E : G  A u is a filter . No tice that E  is a prepa ving containing all si ngletons, and if E contains E , then E  merely coincides with the power set of E . Pr oposition 2.4. Assume th at L is a domain. Let ν be an L -valued maxit ive measur e on E . The map ν  : E  Ñ L defined by (3) ν  p A q  © G P E ,G  A ν p G q is the maximal maxitive measur e extending ν to E  . Pr oof. If ν is defi ned by Equation (2), let I  t denote the collection of a ll A P E  such that A  B for some B P I t . Then p I  t q t P L is a nondecreasing f amily of ideals of E  and, for all A P E  , t t P L : A P I  t u   G P E ,G  A t t P L : G P I t u is a filter i n L . Now the fact that ν  p A q   t t P L : A P I  t u and Proposit ion 2.3 show that ν  is maxitive. The assertion that ν  is the m aximal maxiti ve measure extending ν to E  is not diffi cult and left to the reader .  This corol lary also generalises a result du e to Kramosil [20, Theorem 15.2], where it i s assumed that L is a complete chain (which is necessarily a con tinuous complete semilattice). A proof m ay also be found in [31] in the general s etting of maxitive maps . 3. C A R D I N A L D E N S I T I E S F O R M A X I T I V E M E A S U R E S W e assume in the remaining part of this paper that E is a paving on E , that is a col- lection of subsets of E containing the empty set, closed under finite unions, covering E , and such that, for all x P E , t G P E : G Q x u is nonempty filtered in E (ordered by inclusion). One c ould certainly think of E as the base of some topology G on E . Also, E could be thought of as th e collection of compact subsets of E when equipped wit h some topology O (in which case G coi ncides with the power set of E ), or as the Borel sets of p E , O q . This v ariety of examples e x plains why we do no t assum e E be closed under finite intersections. This also high lights why the hyp othesis E P E , adopted by Akian [3], may be rather restrictive (see the example given abov e, where E is the paving of all finite subsets of E ). The collection of (not necessarily Hausdorff) compact subsets of E for the topology G generated by E i s denoted by K . No te that we a lways ha ve E   K . 4 The fol lo wi ng t heorem gives necessary and suf ficient for a maxitiv e measure to have a dens ity . It goes one step further than [18, Th eore m 3] and [3, Proposit ion 3.15], for we do not ne ed the paving E to be a topology , and the range L of the maxitiv e m easure to be a (locally) complete lattice. Theor em 3.1. Assume that L is a domain, and let ν be an L -valued maxiti ve measur e on E . The follo wing condition s ar e equivalent: (1) ν is completely maxitive, (2) ν is inner-continuous, i.e. for all G P E , ν p G q  à K  G,K P K ν  p K q , (3) ν has a density . If these condi tions are satisfied, ν admits c  : x ÞÑ ν  pt x uq as maxim al density , and c  is an upper semicontinuous map on E . The concept of upper semicontinui ty for po set-v alued m aps, that we do not recall here, is treated by Penot and Théra [29], Beer [7], v an Gool [35], Gerritse [14], Akian and Singer [5]. Pr oof. Fact 1: The restricti on of ν  to K admits c  as cardinal dens ity . Let K P K and m be an upper bound of t c  p x q : x P K u . W e want to show that m ¥ ν  p K q , so let s " m . For any x P K , s " c  p x q   G Q x ν p G q , so there is some G x Q x , G x P E , such that s ¥ ν p G x q . Since K is compact and  x P K G x  K , we can extract a finite subcover and write  k j  1 G x j  K . Thus, s ¥ ν  p K q for any s " m , s o m ¥ ν  p K q thanks to conti nuity of L . Since ν  p K q is i tself an upper boun d of t c  p x q : x P K u , this proves that the supremum of t c  p x q : x P K u exists a nd equals ν  p K q . Fact 2: If either t ν  p K q : K  G, K P K u or t c  p x q : x P G u has a supremum, then à K  G,K P K ν  p K q  à x P G c  p x q . It suffic es to show that both sets have the same upper bo unds. Denoting A Ò for the set of up per bounds o f a subset A  E , the inclusion t ν  p K q : K  G, K P K u Ò  t c  p x q : x P G u Ò is due to th e fa ct that c  p x q  ν  pt x uq and t x u P K for any x P G . The equality holds thanks to Fact 1. Now the impl ications (2) ñ (3) ñ (1) are obvious. Let us s ho w th at (3) ñ (2). Assume t hat ν p G q  À x P G c p x q for all G P E . Then it is easily seen that c can be replaced by c  as a densit y , i .e. ν p G q  À x P G c  p x q for all G P E , and the result can be deduced from Fac t 2. Assume that (1) i s sati sfied and let G P E . An upper bo und of t c  p x q : x P G u is ν p G q . Now let m be an upper bou nd of t c  p x q : x P G u . Let s " m . The definit ion of c  implies that, for all x P G t here is som e G x P E , G x Q x , such that s " ν p G x q . E is a paving, so there is som e H x P E s uch that G X G x  H x Q x . Sin ce G   x P G H x P E 5 and ν is complet ely maxitive, we deduce that s ¥ ν p G q . The conti nuity of L impl ies that m ¥ ν p G q , and (3) is proved. T o conclude, let us show that c  is upper semi-conti nuous, i.e. that t t " c  u is open in E for all t P L . If t " c  p x q , then with the definition of c  there is some G Q x , which i s open in the topology G generated by E , such that t " ν p G q , which implies that G  t t " c  u .  Both forthcoming proposit ions wer e formulated a nd prove d in [2] in the case where E is a top ological space and L i s a continuous lattice, see also [18, Proposition 13]. W e need to cons ider F  t F  E : F P E  and F c P E u and H  K X F . If one takes the case where E is a Hausdorff topology , then F i s the collection of closed subsets, and H  K i s that of compact subsets . Pr oposition 3.2. If L is a do main and ν i s an L -valued maxitive measure on E , then ν pr eserves filter ed intersections of elements of H , i. e . (4) © j P J ν  p H j q  ν  p £ j P J H j q , for every filter ed family p H j q j P J of elements of H such that  j P J H j P H . Pr oof. L et p H j q j P J be a filtered family of elements of H . If all H j are nonempty , then this family has nonempty intersection H . Indeed, if H  H and j 0 P J , then H  H j 0 X  j  j 0 H j , i.e. H j 0   j  j 0 H c j . Since H c j P E , we can extract a finite subcover and write H j 0   k i  1 H c j i , i.e. H  H j 0 X  k i  1 H j i . The family p H j q j P J is filtered, so this implies that one of the H j is empty . Now , let us come back to Equali ty (4). The s et t ν  p H j q : j P J u admits ν  p H q as lower bound. T ake another lower bou nd m , and let G P E such that G  H . The family p H j z G q j P J is a filtered family of elements o f H with em pty intersection, thus H j z G  H for some j P J . Th is implies ν  p H j q ¤ ν p G q , h ence m ¤ ν p G q for all G  H , so th at m ¤ ν  p H q . W e hav e shown that ν  p H q is the g rea stest l o wer bound of t ν  p H j q : j P J u .  T i ghtness for maxitive measures can be defined by analog y with tightness for addi- tiv e measures, so we say that an L -valued maxitiv e measure ν on E is tight if © H P H ν p H c q  0 . (In [32], a tight normed completely maxi ti ve measure on the power set of a topolo g- ical s pace is called a deviability .) If ν is t ight, the collection F can replace H in Proposition 3.2. A semilat tice is a poset in whi ch e very pair t s, t u has a least up per b ound s ` t . A continu ous semil attice is a semilattice which is als o a dom ain. For the fol lo w- ing propo sition we need to recall th at, by [15, Theorem III-2.11], every continuous semilattice L is join-cont inuous in the sense that, for e very t P L and ev ery filter F , t `  F   p t ` F q . 6 Pr oposition 3.3. If L is a continuous semila ttice , and ν i s a tight L -valued maxitive measur e on E , t hen ν pr eserves filter ed i nter sections of elements of F , i.e. © j P J ν  p F j q  ν  p £ j P J F j q , for every filter ed family p F j q j P J of elements of F such that  j P J F j P F . Pr oof. Fix some H P H , and let F   j P J F j . Then F j X H and F X H belong to H , hence  j ν  p F j X H q  ν  p F X H q by Proposition 3.2. Pick some lower bound m of the set t ν  p F j q : j P J u . Thanks to the join-cont inuity of L , m ¤  j p ν  p F j X H q ` ν p H c qq  ν  p F X H q ` ν p H c q ¤ ν  p F q ` ν p H c q . The tight ness of ν and the joi n-continuity of L imply m ¤ ν  p F q , and the result is prov ed.  4. D E C O M P O S I T I O N O F M A X I T I V E M E A S U R E S Here E is again a paving on E . A poset is a lat tice if ev ery finite subset has a supremum and an infimu m. A lattice is di strib utive i f finite i nfima dist rib ute over finite suprema, and locally complet e if e very upp er bounded subset has a supremum. A continuous lo ca lly com plete lattice i s a locally continuous la ttice . A locally continuous lattice whi ch is also di strib u ti ve is a locally cont inuous frame . Note that e very locally continuous frame is a do main. Again R  , R  , and r 0 , 1 s are examples of lo ca lly continuous frames. From Theorem 3.1, the following definition is natural: Definition 4.1. Assum e that L is a locally continuou s lattice, and let ν b e an L -v alued maxitive measure on E . T he r e gular part of ν is the map defined on E by t ν u p G q  à K P K ,K  G ν  p K q . The regular part of ν is a completely (or re gu lar) maxitive measure on E wit h d ensity c  . This is the greatest com pletely maxi ti ve measure lower than ν on E . M oreo ver , t ν u  and ν  coincide on K , hence tt ν uu  t ν u . The follo wi ng th eore m states the e xistence of a r esidu al part K ν of a maxiti ve m ea - sure ν . Theor em 4 .2. Assume t hat L is a locally continuous frame, and let ν be an L -valued maxitive measur e on E . Ther e exists a smal lest maxitive measur e K ν on E , called the r esidual part of ν , such that the decomposition ν  t ν u ` K ν holds. Mor eover , K ν coincides with its own r esi dual part, i .e . KpK ν q  K ν , a nd t he r esidual part of the r e gular part of ν equals 0 , i.e . K t ν u  0 . Pr oof. W e give a constructive proof for the existence of K ν . Let K ν p G q   t t P L : G P I t u , where I t :  t G P E :  H  G, ν p H q ¤ t ν u p H q ` t u . Then p I t q t P L is a nondecreasing family of ideals of E , and distri b utivity imp lies that t t P L : G P I t u 7 is a filter , for e very G P E . From Proposition 2.3, we deduce that K ν is a maxitive measure. Since ν p G q P I t for t  ν p G q , we ha ve ν p G q ¥ K ν p G q , thus ν ¥ t ν u ` K ν . Let us prove t hat the reserve inequality holds. Let G P E , let m be an upper bound of the pair t t ν u p G q , K ν p G qu , and let u " m . There is s ome t P L , ν p G q ¤ t ν u p G q ` t , s uch that u ¥ t . Hence, u ¥ ν p G q , so by continuity of L , m ¥ ν p G q , and the reserve inequality is prove d. T o show that KpK ν q  K ν , first not ice that KpK ν q ¤ K ν , since K ν  t K ν u ` KpK ν q . Second, t K ν u has a density and is lo wer than ν , hence is lower than t ν u . Th us, ν  t ν u ` K ν  t ν u ` t K ν u ` KpK ν q  t ν u ` KpK ν q . This implies that K ν ¤ KpK ν q . The fact that K t ν u  0 is straightforward.  See also [31] for a proof relying on purely order -theoretical p roperties of t he set of maxitive measures. As a consequence of the previous result we have the following corollary . Corollary 4.3. Ass ume that L is a locally continuou s frame, and let ν be an L -valued maxitive measure on E . Then ν has a densit y i f and only if ν  t ν u if and only if K ν  0 . It is worth summarizing ca lculus rules that apply to operators t  u , K , and pq  : Pr oposition 4.4. Assume that L is a locally continuous frame , and let ν, τ P M . Then the following pr operti es hold: (1) ν  t ν u ` K ν , (2) ν  t ν u  K ν  0 , (3) tt ν uu  t ν u , (4) t ν ` τ u  t ν u ` t τ u , (5) p ν ` τ q   ν  ` τ  , (6) KpK ν q  K ν , (7) Kp ν ` τ q ¤ K ν ` K τ , (8) K t ν u  0 . Mor eover , if E is a topology , we have (1) t ν  u  t ν u  , (2) Kp ν  q ¤ pK ν q  . Among the previous l ist one could worry about some desirable property missin g. One naturally expects that the regular part of a residu al part be equal to zero ( t K ν u  0 ), or , i n other words, that the resid ual part to be null on compact subsets, or at least on H . Howe ver , for the latter to be realized we need s ome additional conditions on E , namely that E be closed under the formation of complements ( E is then called a Boolean algebra ). Hence we say t hat a m axiti ve measure ν is s ingular if ν  p H q  0 for all H P H . 8 Theor em 4.5. Assum e t hat E is a Boolean algebra o n E and L is a locally continuous frame. Let ν be an L -valued maxitive measure on E . Then the map defined on E by (5) ν s p G q  © H P H ,H  G ν p G z H q is the gre atest L -valued sin gular maxitive measur e such that the decompo sition ν  t ν u ` ν s holds. Remark 4.6. By Theorem 4.2, K ν is lower than ν s , hence s ingular when E is a Boolean algebra. Pr oof. W e prove that ν s defined by Equation (5 ) is maxiti ve and that the decomposition ν  t ν u ` ν s holds. Let G P E . If H  G , H P H , we hav e, in view of the distributivity of finite joi ns wi th respect to arbitrary meets, ν p G q  ν p H q ` ν p G z H q ¤ t ν u p G q ` ν p G z H q ¤ t ν u p G q ` ν s p G q . The conv erse i nequality is obvious, hence ν  t ν u ` ν s . Next we sho w that ν s is maxitive. If G, G 1 P E , ν s p G q ` ν s p G 1 q  © H  G,H 1  G 1 ν p G z H q ` ν p G 1 z H 1 q ¥ © H  G,H 1  G 1 ν pp G Y G 1 qzp H Y H 1 qq ¥ ν s p G Y G 1 q since H Y H 1 P H , and it remains to show that ν s is nondecreasing. So let G, G 1 P E 1 such that G  G 1 , and let H 1  G 1 , H 1 P H . Th en ν p G 1 z H 1 q ¥ ν p G z H 1 q  ν p G zp G X H 1 qq ¥ ν s p G q , the last inequali ty com ing from t he fa ct that G X H 1 P H since E is stable u nder complementation . W e deduce that ν s p G 1 q ¥ ν s p G q , and ν s is maxitive. Also, ν s is singular since, for all H P H , ν  s p H q  ν s p H q  0 . Suppose that ν  t ν u ` τ for some singular maxit i ve measure τ , and let G P E . Then, for all H  G , H P H , τ p G q  τ p G z H q ` τ p H q  τ p G z H q by singularity of τ . Hence τ p G q ¤ ν p G z H q , and we get τ p G q ¤ ν s p G q for all G P E .  Corollary 4.7. If E is a Boolean algebra on E and L is a local ly continuous frame, then eve ry L -valued tigh t maxitive measur e on E has a density . Pr oof. L et ν be a tight maxitive measure on E . Since K ν ¤ ν s , we hav e in particular , for all G P G , K ν p G q ¤ ν s p E q   H P H ν p H c q  0 . Thi s means that the singular part of ν is null, so that ν  t ν u , and ν has a cardinal density thanks to Proposition 3.1.  Acknowledgeme nts. I w ould li k e to thank Marianne Akian for her valuable comments and suggestions . R E F E R E N C E S [1] Emilio Acerbi, Giuseppe Buttazzo, and France sca Prinari. The class of fun ctionals which can be represented by a supremu m. J. Con vex Anal. , 9(1):225 –236, 2002. 9 [2] Mariann e Ak ian. Theo ry of cost mea sures: conv ergence o f decision variables. 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C M A P , É C O L E P O L Y T E C H N I Q U E , R O U T E D E S A C L A Y , 9 1 1 2 8 P A L A I S E AU C E D E X , F R A N C E , A N D I N R I A , S AC L A Y – Î L E - D E - F R A N C E E-mail addr ess : poncet@cmap. polytechnique.fr 11