International Journal of Analysis and Applications ISSN 2291-8639 Volume 9, Number 2 (2015), 114-120 http://www.etamaths.com ON THE INTEGRAL REPRESENTATION OF STRICTLY CONTINUOUS SET-VALUED MAPS ANATÉ K. LAKMON∗ AND KENNY K. SIGGINI Abstract. Let T be a completely regular topological space and C(T ) be the space of bounded, continuous real-valued functions on T . C(T ) is endowed with the strict topology (the topology generated by seminorms determined by continuous functions vanishing at infinity). R. Giles ([13], p. 472, Theorem 4.6) proved in 1971 that the dual of C(T ) can be identified with the space of regular Borel measures on T . We prove this result for positive, additive set- valued maps with values in the space of convex weakly compact non-empty subsets of a Banach space and we deduce from this result the theorem of R. Giles ([13], theorem 4.6, p.473). 1. Introduction The strict topology β was for the first time introduced by R. C. Buck ([1], [2]) on the space C(T) of all bounded continuous functions on a locally compact space T. He has proved among others that the dual space of (C(T),β) is the space of all finite signed regular Borel measures on T . After a large number of papers have appeared in the literature concerned with extending the results contained in Buck’s paper [1]( see e.g. [4], [5], [6], [7], [8], [12],[14], [15], [17], [18], [19], [22], [25] and [27]). R. Giles has generalized this notion of the strict topology introduced by Buck for completely regular space T and has proved Buck’s results, particulary the theorem 2 in [1] for an arbitrary (not necessarily Hausdorff) completely regular space T. In this paper we generalize Giles’s result ([13], theorem 4.6, p.473) to additive, positive, positively homogeneous and strictly continuous set-valued maps defined on C+(T) with values in the space cc(E) of all convex weakly compact non-empty subsets of a Banach space E. We deduce from this result the theorem of R. Giles. 2. Notations and definitions Let T be a completely regular topological space and let B(T) be the Borel σ- algebra of T and let C(T) be the space of bounded continuous real-valued functions on T . Let C0(T) be the subspace of C(T) consisting of functions f vanishing at infinity i.e. for any ε > 0 there is a compact set Kε ⊂ T such that |f(x)| < ε for x ∈ T\Kε. We denote by C+(T) the subspace of C(T) consisting of non-negative functions and by 1A the characteristic function of each A ⊂ T . For all f ∈ C(T), we put f+ = sup(f, 0),f− = sup(−f, 0) and ||f||∞ = sup{|f(t)|; t ∈ T}. We denote 2010 Mathematics Subject Classification. Primary 28B20, Secondary 54C60. Key words and phrases. set-valued measure; strict topology; regular set-valued measure; ad- ditive and positive set-valued map. c©2015 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 114 ON THE INTEGRAL REPRESENTATION 115 by R the set of real numbers. Let E be a Banach space, E′ its dual and cc(E) be the space of all non-empty, convex weakly compact subsets of E; we denote by ‖.‖ the norm on E and E′. If X and Y are subsets of E we shall denote by X +Y the family of all elements of the form x+y with x ∈ X and y ∈ Y . The support function of X is the function δ∗(.|X) from E′ to [−∞; +∞] defined by δ∗(y|X) = sup{y(x),x ∈ X}. We endow cc(E) with a Hausdorff distance, denoted by δ. For all K ∈ cc(E) and for all K′ ∈ cc(E),δ(K,K′) = sup{|δ∗(y|K) − δ∗(y|K′)|; y ∈ E′,‖y‖≤ 1}. Recall that (cc(E),δ) is a complete metric space ([16], theorem 9, p.185) and ([21], theorem 15, p.2-2). Definition 2.1. (1) let m : B(T) → R be a positive countable additive measure. We say that m is: (i) inner regular if for all A ∈B(T) and ε > 0, there exists a compact Kε subset of T such that Kε ⊂ A and m(A\Kε) < ε. (ii) outer regular if for all A ∈B(T) and for all ε > 0, there exists an open subset Oε of T such that Oε ⊃ A and m(Oε\A) < ε. (iii) regular if it is inner regular and outer regular. (2) A signed measure µ : B(T) → R is regular if and only if its total variation v(µ) is regular. Note that v(µ) : B(T) → R+ (A 7→ v(µ)(A) = sup{ ∑ i |µ(Ai)|; (Ai) finite partition of A,Ai ∈B(T)}). Definition 2.2. A map M : B(T) → cc(E) is a set-valued measure if M(A ∪ B) = M(A) + M(B) for every pair of disjoint sets A,B in B(T),M(∅) = {0} and M( +∞⋃ n=1 An) = +∞∑ n=1 M(An) for every sequence (An) of mutually disjoint elements of B(T); which amounts to saying that for all y ∈ E′ the map δ∗(y|M(.)) : B(T) → R(A 7→ δ∗(y|M(A))) is a countably additive measure ([21], corollary p. 2-25). We say that a set-valued measure M is: (i) positive if for all A ∈B(T), 0 ∈ M(A) (ii) regular if for all y ∈ E′, the measure δ∗(y|M(.)) is regular. Let ϕ ∈ C0(T), let K be a compact subset of T . We denote by pϕ and pK the semi-norms on C(T) defined by pϕ(f) = sup{|f(t)ϕ(t)|; t ∈ T} and pK(f) = sup{|f(t)|; t ∈ K} for every f ∈ C(T). Definition 2.3. The topology determined by the set of semi-norms {pϕ; ϕ ∈ C0(T)} (resp. {pK; K belongs to the family of compact subsets of T}) is called the strict (resp. the compact convergence) topology. We say that a map defined on C(T) is strictly continuous if it is continuous for this topology. Definition 2.4. A map L : C+(T) → cc(E) is: (i) additive set-valued map if for all f,g ∈ C+(T) L(f + g) = L(f) + L(g) (ii) positively homogeneous if for f ∈ C+(T) and for λ ≥ 0 L(λf) = λL(f). (iii) positive if for every f ∈ C+(T), 0 ∈ L(f). Definition 2.5. ([24], p. 04) Let m be a bounded linear functional on C(T), and let B(0, 1) be the unit ball of C(T). We say that m is tight if its restriction to B(0, 1) is continuous for the topology of compact convergence. 116 LAKMON AND SIGGINI 3. Main result Lemma 3.1. Let m be a bounded linear functional on C(T). If m is tight then for all ε > 0 there is a compact subset Kε of T such that for all f ∈ C(T) and |f| ≤ 1T\Kε , we have |m(f)| < ε. Proof. Assume that m is tight. Then for every ε > 0 there is a compact subset Kε of T and there is η > 0 such that for all f ∈ B(0, 1) and pKε (f) = sup{|f(t)|; t ∈ Kε} < η. We have |m(f)| < ε. In particular for all f ∈ B(0, 1) such that |f| ≤ 1T\Kε , one has |m(f)| < ε. � Lemma 3.2. Let M : B(T) −→ cc(E) be a positive, regular set-valued measure. Then the real-valued measure δ∗(y|M(.)) are uniformly tight with respect to y ∈ E′,‖y‖ ≤ 1 ie for every A ∈ B(T) and for every ε > 0 there is a compact subset Kε of T such that Kε ⊂ A and sup{δ∗(y|M(A\Kε)); y ∈ E′,‖y‖≤ 1}≤ ε. Proof. Let us consider the set {δ∗(y|M(.)),y ∈ E′,‖y‖ ≤ 1} of countably additive real-valued measures. It is uniformly countable additive (see [9], theorem 10, p. 88–89; [28], lemma 3.1, p. 275). According to ([10], p. 443, Theorem 10.7) there is a sequence (cn) of real numbers and there is a sequence (δ ∗(yn|M(.))), |yn| ≤ 1 of measures such that µ(A) = +∞∑ n=1 cnδ ∗(yn|M(A)) exists for each A ∈ B(T) and such that the series ∑ |cn|δ∗(yn|M(A)) is uniformly convergent for A ∈B(T); moreover the countable additive measure ν : B(T) → R(A 7→ ν(A) = +∞∑ n=1 |cn|δ∗(yn|M(A))) verifies the following relation: lim ν(A)→0 [sup{δ∗(y|M(A)); y ∈ E′,‖y‖ ≤ 1}] = 0 (*). We deduce from the uniform convergence of the series ∑ |cn|δ∗(yn|M(A)) for A ∈ B(T), that ν is regular. Indeed, given ε > 0 choose n0 ∈ N such that sup A∈B(T) ∣∣∣∣ν(A) − n0∑ k=1 |ck|δ∗(yk|M(A)) ∣∣∣∣ < ε/2. For A ∈ B(T), choose a compact subset K of T such that K ⊂ A and for every k ∈ {1, 2, ...,n0} δ∗(yk|M(A\K)) ≤ ε2(n0+1)r0 with r0 = sup{|ck|; k ∈ {1, 2, ...,n0}} then n0∑ k=1 |ck|δ∗(yk|M(A\K)) ≤ ε/2, therefore ν(A\K) ≤ ε. The relation (*) and the inner regularity of ν show that for each ε > 0 and each A ∈ B(T) there exists a compact subset K of T such that K ⊂ A and sup{δ∗(y|M(A\K)); y ∈ E′,‖y‖≤ 1}≤ ε. � Let M be a positive set-valued measure defined on B(T). For the construction of the integral ∫ fM, with f ∈ C+(T) we refer to ([23], p. 17). Lemma 3.3. Let M : B(T) → cc(E) be a positive regular set-valued measure. Then the set-valued map L : C+(T) → cc(E)(f 7→ L(f) = ∫ fM) is additive, positively homogeneous, positive and strictly continuous. Proof. We only prove the strict continuity. The other properties follow from the construction of the integral ∫ fM,f ∈ C+(T). For each n ∈ N∗ there exists a compact subset Kn of T such that sup{δ∗(y|M(T\Kn)); y ∈ E′,‖y‖ ≤ 1} ≤ 2−2n (Lemma 3.2). We then have a sequence (Kn) of compact subsets of T that we may assume monotone increasing. We repeat here the proof of R. Giles ([13], p. 471, ON THE INTEGRAL REPRESENTATION 117 Lemma 4.2). Consider ϕ = +∞∑ n=1 2−n1Kn , we have 2 −n−1 ≤ ϕ(x) ≤ 2−n for all x ∈ Kn+1\Kn. The function 1/ϕ is measurable and is δ∗(y|M(.)) - integrable for each y ∈ E′,‖y‖ ≤ 1. We have ∫ 1/ϕδ∗(y|M(.)) = ∫ ∪+∞n=1(Kn+1\Kn) 1/ϕ δ∗(y|M(.)) = +∞∑ n=1 ∫ Kn+1\Kn 1/ϕ δ∗(y|M(.)) ≤ +∞∑ n=1 2n+1 [δ∗(y|M(Kn+1)) −δ∗(y|M(Kn))] ≤ +∞∑ n=1 2n+1.2−2n = 2. Let ε > 0 and let ψn ∈ C0 such that ψn(x) = 2−n for x ∈ Kn and 0 ≤ ψn ≤ 2−n1T . Put ψ = +∞∑ n=1 ψn. Then ψ ∈ C0 and ϕ ≤ ψ. For all f ∈ {g ∈ C+(T),p2ψ/ε(g) < 1} we have f < ε/2ϕ and ∫ fδ∗(y|M(.)) < ε for all y ∈ E′ with ‖y‖ ≤ 1. Since δ∗(y| ∫ fM) = ∫ fδ∗(y|M(.)), one has δ( ∫ fM,{0}) < ε. Therefore the map f →∫ fM is strictly continuous at 0. The equality δ∗(y| ∫ fM) = ∫ fδ∗(y|M(.)) for each f ∈ C+(T) and each y ∈ E′ enable us to prove the continuity on C+(T). � Definition 3.4. A map S : E′ → R is said to be sublinear if for every y ∈ E′ and y′ ∈ E′ and for every λ ≥ 0 one has S(y + y′) ≤ S(y) + S(y′) and S(λy) = λS(y). The lemme below is a particular case of L. Hörmander’s result ([16], Theorem 5, p. 182). We give here an alternative proof. Lemma 3.5. Let E be a Banach space, and let E′ its dual space endowed with the Mackey topology τ(E′,E). Let S : E′ → R be a sublinear map. Then S is continuous if and only if there is C ∈ cc(E) such that S = δ∗(.|C). Proof. Assume that S is continuous. Let ∇S = {l : E′ → R; linear and l ≤ S}. By the Hahn-Banach theorem ([11], theorem 10, p. 62), S(y) = sup{l(y); l ∈∇S} for each y ∈ E′. Let l ∈∇S; then l is continuous for the Mackey topology τ(E′,E). Therefore l determines an element xl ∈ E that verifies l(y) = y(xl) for each y ∈ E′. Let ∇ES = {xl; l ∈ ∇S}. Since ∇S is equicontinuous there is a neighborhood V of 0 in E′ such that ∇ES ⊂ V ◦, where V ◦ is the polar of V in E. By the Alaoglu-Bourbaki’s theorem ([20], p. 248), one has V ◦ ∈ cc(E). Since ∇ES is convex , its closure is one of elements of cc(E) we want. The converse is obvious. Note that if S is non-negative then 0 ∈∇ES. � Theorem 3.6. Let T be a completely regular topological space and let C+(T) be the space of bounded continuous non-negative functions defined on T endowed with the strict topology. Let E be a Banach space and cc(E) be the space of convex weakly compact non-empty subsets of E endowed with the Hausdorff distance. Let L : C+(T) → cc(E) be a positive, additive, positively homogeneous and strictly continuous set-valued map. Then there is a unique positive regular set-valued mea- sure M defined on B(T) to cc(E) such that L(f) = ∫ fM for all f ∈ C+(T). Conversely for all positive regular set-valued measure M : B(T) → cc(E), the set- valued map θ : C+(T) → cc(E) (f 7→ θ(f) = ∫ fM) is positive, additive, positively homogeneous and strictly continuous. Proof. Let y ∈ E′. The map δ∗(y|L(.)) : C+(T) → R (f 7→ δ∗(y|L(f))) is additive, positively homogeneous and continuous. Then it can be extended to a continuous linear functional on C(T). This extension is unique. It is denoted by δ∗(y|L̄(.)). Let f ∈ C(T), one has f = f+ − f− and δ∗(y|L̄(.)) is defined by δ∗(y|L̄(.))(f) = 118 LAKMON AND SIGGINI δ∗(y|L(f+)) − δ∗(y|L(f−)). Since δ∗(y|L̄(.)) is strictly continuous it is tight ([26], p. 41). By the lemma 3.1 and ([3], Proposition 5, p.58) there exists a unique regular positive Borel measure µy on T that verifies δ ∗(y|L̄(f)) = ∫ fµy for all f ∈ C(T). Let 0 an open subset of T and let SO the map defined on E′ to R by SO(y) = µy(O) for each y ∈ E′. We have µy(O) = sup{ ∫ fµy; f ∈ C+(T),f ≤ 1O} = sup{δ∗(y|L(f)); f ∈ C+(T),f ≤ 1O}, therefore SO is a sublinear map. Let now A ∈B(T). We denote by SA the map defined on E′ to R by SA(y) = µy(A) for each y ∈ E′. Since the measure µy is regular we have SA(y) = inf{µy(O); O ⊂ T,O open and O ⊃ A} = inf{SO(y); O ⊂ T,O open and O ⊃ A}. Let y,y′ ∈ E′ and let ε > 0, there exists two open subsets Oε and O ′ ε of T containing A and such that SA(y) ≥ µy(Oε) −ε/2, SA(y′) ≥ µy′ (O′ε) −ε/2. We have µy(Oε) + µy′ (O′ε) ≤ SA(y) +SA(y ′) +ε, then µy(Oε∩O′ε) +µy′ (Oε∩O′ε) ≤ SA(y) +SA(y′) +ε, therefore µy+y′ (Oε ∩ O′ε) ≤ SA(y) + SA(y′) + ε. We have µy+y′ (A) ≤ µy+y′ (Oε ∩ O′ε) ≤ SA(y) + SA(y ′) + ε. It follows from this SA(y + y ′) ≤ SA(y) + SA(y′). It is obvious that for all λ ≥ 0 and for all y ∈ E′, SA(λy) = λSA(y). So SA is a non- negative sublinear map. Let us prove now that SA is continuous for the Mackey topology τ(E′,E). We have SA(y) ≤ µy(T) = δ∗(y|L(1T )). Let L̃(1T ) be the closed absolutely convex cover of L(1T ), one has L̃(1T ) ∈ cc(E) and SA(y) ≤ δ∗ ( y|L̃(1T ) ) for each y ∈ E′ and A ∈ B(T). We deduce that SA is continuous for the Mackey topology for each A ∈ B(T). By the lemma 3.5 there is CA ∈ cc(E) such that SA(y) = δ ∗(y|CA) for all y ∈ E′. Let M : B(T) → cc(E) (A 7→ M(A) = CA). We have δ∗(y|M(A)) = µy(A) for all y ∈ E′, hence the map δ∗(y|M(.)) : B(T) → R (A 7→ δ∗(y|M(A))) is a positive regular countably additive measure. Then M is a regular set-valued measure. Since SA is non-negative then M is positive. Let f ∈ C+(T) and let y ∈ E′, ∫ fδ∗(y|M(.)) = ∫ fµy = δ ∗(y|L(f)). It follows that L(f) = ∫ fM for all f ∈ C+(T) because ∫ fδ∗(y|M(.)) = δ∗(y| ∫ fM). Let us prove that M is unique. Assume that there exist two regular positive set-valued measures M and M′ which verify ∫ fM = L(f) = ∫ fM′. Let 0 be an open subset of T and let y ∈ E′. According to the inner regularity of δ∗(y|M(.)) and ([3] Lemme 1 p. 55) we have δ∗(y|M(O)) = sup{δ∗(y|L(f)); f ∈ C+(T), f ≤ 1O} = δ∗(y|M′(O)). Moreover the outer regularity of δ∗(y|M(.)) shows that δ∗(y|M(A)) = δ∗(y|M′(A)) for all A ∈ B(T) and y ∈ E′, hence M(A) = M′(A) for all A ∈ B(T). The second assertion of the theorem is justified by the lemma 3.3. � The following corollary is the result of R. Giles. Corollary 3.7. ([13], Theorem 4.6 ) For any completely regular space T the dual of C(T) under the strict topology is the space of all bounded signed Borel regular measures on T . Proof. Let L be a strictly continuous linear functional on C(T); L is bounded. Therefore L is the difference of two non-negative linear functional. We may assume that L is non-negative. Let K0 be an element of cc(E) that contains 0 and that is subset of the unit ball of E. Consider the map L′ : C+(T) → cc(E) defined by L′(f) = L(f)K0 = {L(f)k; k ∈ K0} for all f ∈ C+(T). The map L′ is positive, positively homogeneous and strictly continuous. Let us prove that L′ is additive. The inclusion L′(f + g) ⊂ L′(f) + L′(g) for all f,g ∈ C+(T) is trivial. Let u ∈ K0 and each let v ∈ K0, L(f)u + L(g)v = L(f + g) [ L(f) L(f+g) u + L(g) L(f+g) v ] . Since K0 is convex and L positive, L(f) L(f+g) u + L(g) L(f+g) v ∈ K0. Then L′(f) + L′(g) ⊂ L′(f + g). ON THE INTEGRAL REPRESENTATION 119 By the Theorem 3.6, there is a unique positive regular set-valued measure M : B(T) → cc(E) that satisfies the condition ∫ fM = L′(f) for all f ∈ C+(T). Let y0 ∈ E′ such that δ∗(y0|L′(.)) = L. Since δ∗(y0| ∫ fM) = ∫ fδ∗(y0|M(.)) for all f ∈ C+(T) we then have ∫ fδ∗(y0|M(.)) = L(f) for all f ∈ C+(T) and therefore∫ fδ∗(y0|M(.)) = L(f) for all f ∈ C(T). The uniqueness of δ∗(y0|M(.)) follows from the regularity of M. Taking the lemma 3.3 (for the scalar measures) into account we conclude that there is a bijection between the dual space of (C(T),β) and the space of all bounded signed regular Borel measures on T . � References [1] R. C. Buck, Bounded continuous functions on locally compact space, Michigan Math. J. 5 (1958), 95–104. [2] R. C. Buck, operator algebras and dual spaces, Proc. Amer. Math. Soc. 3.681– 687 (1952). [3] BOURBAKI, Eléments de Maths. Livre VI Intégration-Chp. IX, Ed. Hermann, Paris 1969. [4] A. Choo, strict topology on spaces of continuous vector-valued functions, Canad. J. Math. 31 (1979), 890–896. [5] A. Choo, Separability in the strict topology, J. Math. Anal. Appl. 75 (1980), 219–222. [6] H. S Collins, On the space l∞(S), with the stict topology, Math. Zeitschr. 106, 361–373 (1968). [7] H. S. Collins and J. R. Dorroh, Remarks on certain function spaces, Math. Ann., 176, (1968), 157–168 . [8] J. B. Conway, the strict topology and compactness in the space of measures, Bull. Amer. Math. Soc. 72, (1966), 75–78 . [9] , J. Diestel, Sequences and Series in Banach spaces, Graduate Texts in Math., vol.92, Springer-Verlag, 1984. [10] Drewnowsky, Topological Rings of Sets, Continuous Set Functions, Integra- tion. III, Bull. Acad. Polon. Sci., Sér. Sci. Math., Astronom. et Phys., 20 (1972), 441–445. [11] N. Dunford and J. Schwartz, Linear operators Part I, New York: Interscience 1958. [12] R. A. Fontenot, Strict topologies for vector-valued functions, Canadian. J. Math. 26 (1974), 841–853. [13] R. Giles, A Generalization of the Strict Topology, Trans. Amer. Math. Soc. 161(1971), 467–474. 120 LAKMON AND SIGGINI [14] D. Gulick, The σ-compact-open topology and its relatives, Math. Scand.. 30 (1972), 159–176. [15] J. Hoffman-Jörgenson, A generalization of strict topology, Math. Scand. 30 (1972), 313–323. [16] L. Hörmander, Sur la fonction d’appui des ensembles convexes dans un espace localement convexe, Arkiv För MATEMATIK. 3 nr 12 (1954). [17] A. K. Katsaras, On the strict topology in the non-locally convex setting II, Acta. Math. Hung. 41 (1-2) (1983), 77–88. [18] A. K. Katsaras, Some locally convex spaces of continuous vector-valued func- tions over a completely regular space and their duals, Trans. Amer. Math. Soc. 216 (1979), 367–387. [19] L. A. Khan, The strict topology on a space of vector-valued functions, Proc. Edinburgh Math. Soc., 22 (1979), 35–41. [20] G. Köthe, Topological Vector spaces I Second printing, Springer-Verlag, New- York, 1983. [21] R. Pallu De La Barriere, Publications Mathématiques de l’Université Pierre et Marie Curie No33. [22] F. D. Sentilles, Bounded continuous functions on a completely regular space, Trans. Amer. Math. Soc., 168 (1972), 311–336. [23] K. K. Siggini, Narrow Convergence in Spaces of Set-valued Measures, Bull. of The Polish Acad. of Sc. Math. Vol. 56, N◦1, (2008). [24] K. K. Siggini, Sur les propriétés de régularité des mesures vectorielles et multi- voques sur les espaces topologiques généraux, Thèse de Doctorat de l’Université de Paris 6. [25] C. Todd, Stone-Weierstrass theorems for the strict topolgy, Proc. Amer. Math. Soc. 16 (1965), 657–659. [26] A. C. M. Van Rooij, Tight functionals and the strict topology, Kyungpook Math.J.7 (1967), 41–43. [27] J. Wells, Bounded continuous vector-valued functions on a locally compact space, Michigan Math. J. 12 (1965), 119–126. [28] X. Xiaoping, C. Lixin, L. Goucheng, Y. Xiaobo, Set valued measures and integral representation, Comment.Math.Univ.Carolin. 37,2 (1996)269–284 University of Lomé, Faculty of Sciences, Department of mathematics, BP 1515 Lomé- TOGO ∗Corresponding author