Radon-Nikodym representations for multivalued Bartle-Dunford-Schwartz integral
aa r X i v : . [ m a t h . F A ] F e b RADON-NIKOD ´YM REPRESENTATIONS FOR MULTIVALUEDBARTLE-DUNFORD-SCHWARTZ INTEGRAL
LUISA DI PIAZZA, KAZIMIERZ MUSIA L, ANNA RITA SAMBUCINI
Abstract.
An integral for a scalar function with respect to a multimeasure N taking its values in alocally convex space is introduced. The definition is independent of the selections of N and is relatedto a functional version of the Bartle-Dunford-Schwartz integral with respect to a vector measurepresented by Lewis. Its properties are studied together with its application to Radon-Nikod´ymtheorems in order to represent as an integrable derivative the ratio of two general multimeasures ortwo d H -multimeasures; equivalent conditions are provided in both cases. Introduction
The theory of the Bartle-Dunford-Schwartz-integral (BDS-integral) of a scalar function with respectto a vector valued measure was introduced in 1955 by R. C. Bartle, N. Dunford and J. T. Schwartz [1],and subsequently extensively studied by several authors ( [13, 18–20, 22, 27, 29, 30]). In particular in‘70 Lewis [18] proposed an equivalent functional version of it (see also [19, 20, 27]). The literatureconcerning the Bartle-Dunford-Schwartz integration is rather wide so we quote here only those paperswhich are close to the topic of our work and two books ( [20, 27]). We would like however to payattention to [12] and the references therein, where a kind of the BDS integral with respect to measureswith values in a sequentially complete linear topological space is investigated.In [22] the second author proved the existence of the Radon-Nikod´ym derivative of a vector valuedmeasure ν with respect to a vector valued measure κ by means of the BDS-integral, under suitablehypotheses on the measures ν and κ . From this article then came out the versions with respectto finitely additive measures [21], that of [5] where a Radon-Nikod´ym density is obtained in themultivalued case via the Gould integral and that of [4], where the derivative belongs to a suitablespace. We recommend [8, 26] for readers interested in Radon-Nikod´ym results in the context of Pettisintegration.In [17] Kandilakis defined an integral of a scalar function with respect to a multimeasure N , whosevalues were weakly compact and convex subsets of a Banach space. This integral is constructed usingthe selections of N . Contrary to the classical BDS-integral, the Kandilakis’ integral is only sublinearwith respect to the integrable functions.It is our aim to define an integral with respect to a multimeasure N , independent of the selections of N . To achieve it we take into account the support functions of N . Our approach is a consequenceof Kandilakis’ calculations. The considered multimeasures N are very general, in fact they can takenonempty closed, convex values in an arbitrary locally convex space X . Sometimes we will assumequasi-completeness of X (closed bounded sets are complete).In Section 2 we study properties of the integral. Moreover, we compare the Aumann-BDS integraland the new one: if the multimeasure N possesses selections and the scalar integrable function isbounded, then the Aumann-BDS integrability implies the N -integrability. To obtain the main resultswe require the existence of control measures for the multimeasures under investigation and in Section3 we give conditions guaranteeing the existence of such controls.As an application of the new integral, we study the problem of existence of the Radon-Nikod´ym deriv-ative of a given multimeasure M with respect to a multimeasure N . In the classical measure theory, Mathematics Subject Classification.
Primary 28B20; Secondary 26E25, 26A39, 28B05, 46G10, 54C60, 54C65.
Key words and phrases.
Locally convex space, multifunction, Bartle-Dunford-Schwartz integral, support function,selection, Radon-Nikod´ym theorem. the Radon-Nikod´ym theorem states in concise conditions, namely absolute continuity, domination andsubordination, how a measure can be factorized by another measure through a density function.In Section 4 we first consider arbitrary multimeasures. In such a case we are able to characterizethe existence of the Radon-Nikod´ym derivative of M with respect to N (see Theorem 4.4) by meansthe notions of uniform scalar absolutely continuity, uniform scalar domination and uniform scalarsubordination.In Section 5 multimeasures which are countably additive in the Hausdorff metric are studied. In sucha case the multimeasures take values in nonempty, bounded, closed, convex subsets of a locally convexspace. The differentiation of M with respect to N is in general not equivalent to differentiation of itsR˚adstr¨om embedding j ◦ M with respect to j ◦ N . The reason is that one has to take into accountalso integration with respect to the measure j ◦ ( − N ). In the case of multimeasures, we have ingeneral j ◦ ( − N ) = − j ◦ N . The main result is Theorem 5.6 where we find conditions (the stronguniform scalar absolutely continuity, the strong uniform scalar domination and the strong uniformscalar subordination) guaranteing the existence of the Radon-Nikod´ym derivative both of M withrespect to N and of their R˚adstr¨om embeddings.At last in Section 6 we provide some examples of multimeasures that can be represented by anothermultimeasure through a multivalued Bartle-Dunford-Schwartz density.2. Preliminaries
Throughout (
Ω, Σ ) is a measurable space, the real numbers are denoted by R and R +0 denotes thenon-negative reals. If ν : Σ → ( −∞ , + ∞ ] is a measure, then | ν | denotes its variation. Σ E is the familyof all Σ -measurable subsets of E . Let X be a locally convex linear topological space (shortly, locallyconvex space) and let X ′ be its conjugate space. Given a subset S of X , we write co( S ), aco( S ) andspan( S ) to denote, respectively, the convex, absolutely convex and linear hull of S .The symbol c ( X ) denotes the collection of all nonempty closed convex subsets of X and cb ( X ) , cwk ( X ), ck ( X ) denote respectively the family of all bounded and the family of all (weakly) compact membersof c ( X ). For every C ∈ c ( X ) the support function of C is denoted by s ( · , C ) and defined on X ′ by s ( x ′ , C ) = sup {h x ′ , x i : x ∈ C } , for each x ′ ∈ X ′ . The symbol ⊕ denotes the closure of the Minkowskiaddition.We say that M : Σ → c ( X ) is a multimeasure if for every x ′ ∈ X ′ the set function s ( x ′ , M ( · )) : Σ → ( −∞ , + ∞ ] is a σ -finite measure. The σ -finiteness of each s ( x ′ , M ) seems to be the weakestpossible assumption, otherwise we meet problems connected with the RN-theorem. Simply if some ν := s ( x ′ , M ) is not σ -finite but is absolutely continuous with respect to a finite measure µ , then thereis a set Ω such that ν is σ -finite on Ω and takes only values 0 , + ∞ on Ω c . If ν ( E ) = Z E f dµ , then f must take infinite values and we are not interested in such a situation.Given a multimeasure M : Σ → c ( X ) we denote by N ( M ) := { E ∈ Σ : M ( E ) = { }} the familyof null sets. A multimeasure M : Σ → c ( X ) is said to be σ - bounded if there is a sequence ( Ω n ) n ofelements of Σ such that Ω \ S n Ω n ∈ N ( M ) and M is cb ( X )-valued on each algebra Σ Ω n .A multimeasure M is called pointless if its restriction to no set E ∈ Σ \ N ( M ) is a vector measure.Each multimeasure determined by a function (see [6] for the definition) that is not scalarly equivalent tozero function is pointless. Less trivial examples can be deduced from [24, Example 1.11] if one assumesthat the function r appearing there is strictly positive. Two multimeasures M, N : Σ → c ( X ) are consistent if there exists H ∈ Σ such that M and N are pointless on H and vector measures on H c .If M is a cb ( X )-valued multimeasure, then for each x ′ ∈ X ′ the measure s ( x ′ , M ) is finite. If A ∈ Σ , then M | A is the multimeasure defined on Σ A by ( M | A )( E ) := M ( A ∩ E ). A multimea-sure M : Σ → c ( X ) is called positive , if 0 ∈ M ( E ) for each E ∈ Σ .If X is a Banach space and a multimeasure M : Σ → cb ( X ) is countably additive in the Hausdorffmetric d H , then it is called a d H - multimeasure . ADON-NIKOD´YM REPRESENTATIONS FOR THE
BDS m -INTEGRAL 3 In the following if Z is any metric space, we use the symbol B Z to denote its closed unit ball.A helpful tool to study the d H -multimeasures is the R˚adstr¨om embedding j : cb ( X ) → ℓ ∞ ( B X ′ ),defined by j ( A ) := s ( · , A ), (see, for example [2, Theorem 3.2.9 and Theorem 3.2.4(1)] or [9, TheoremII-19]) It is known that B X ′ can be embedded into ℓ ′∞ ( B X ′ ) by the mapping x ′ −→ e x ′ , where h e x ′ , h i = h ( x ′ ), for each h ∈ ℓ ∞ ( B X ′ ). Moreover, the range of B X ′ is a norming subset of ℓ ′∞ ( B X ′ ).The embedding j satisfies the following properties: j ( αA ⊕ βC ) = αj ( A ) + βj ( C ) for every A, C ∈ cb ( X ) , α, β ∈ R + ; d H ( A, C ) = k j ( A ) − j ( C ) k ∞ , A, C ∈ cb ( X ); j ( cb ( X )) is a closed cone in the space ℓ ∞ ( B X ′ ) equipped with the norm of the uniformconvergence.Observe that instead of ℓ ∞ ( B X ′ ), we may use C B ( B X ′ , τ ( X ′ ,X ) ) (where τ ( X ′ ,X ) is the Mackey topology)for weakly compact sets, C ( B X ′ , σ ( X ′ , X )) in case of compact sets and C B ( B X ′ , k · k ∞ ) in case ofclosed bounded sets. But as we do not apply any special properties of space in which we embed cb ( X ),we will stay with ℓ ∞ ( B X ′ ) . (1)We denote by R ( M E ) the range of the multimeasure M in c ( X ), restricted to measurable subsets of E : R ( M E ) := { M ( F ) : F ∈ Σ E } ⊂ c ( X ). Moreover, R ( M E ) := { z ∈ X : ∃ F ∈ Σ E , z ∈ M ( F ) } ⊂ X .We say that a multimeasure M : Σ → c ( X ) is absolutely continuous with respect to a multimeasure N : Σ → c ( X ) (we write then M ≪ N ), if N ( N ) ⊂ N ( M ) . If M ≪ N and N ≪ M , then themultimeasures are called equivalent.We say that a non negative measure µ is a control measure for a multimeasure M : Σ → c ( X ) if µ is a finite measure and for each E ∈ Σ the condition µ ( E ) = 0 yields M ( E ) = { } . If µ is acontrol measure for M : Σ → c ( X ), then M is σ -bounded if and only if it is locally bounded, i.e. foreach set E / ∈ N ( µ ) there exists a subset F of E of positive µ -measure such that M is cb ( X )-valuedon Σ F . It is a direct consequence of [1, Theorem 1.4] that each d H -multimeasure has a control measure.If M : Σ → c ( X ) is a multimeasure, then S M denotes the family of all countably additive X -valuedselections of M . Definition 2.1. • Let n : Σ → X be a vector measure. A measurable function f : Ω → R is called Bartle-Dunford-Schwartz (BDS) integrable with respect to n , if for every x ′ ∈ X ′ f is h x ′ , n i -integrableand for each E ∈ Σ there exists a point ν ( E ) ∈ X such that h x ′ , ν ( E ) i = Z E f d h x ′ , n i , forevery x ′ ∈ X ′ . Such an approach to the Bartle-Dunford-Schwartz integral was suggested byLewis [18]. • Let N : Σ → c ( X ) be a multimeasure. A measurable function f : Ω → R is called Aumann-Bartle-Dunford-Schwartz (Aumann-BDS)-integrable with respect to N if S N = ∅ and f isintegrable in the sense of Bartle-Dunford-Schwartz with respect to all members of S N . Theintegral on a set E ∈ Σ is then defined by the formula( s ) Z E f dN := (cid:26) ( BDS ) Z E f dn : n ∈ S N (cid:27) . The above definition was suggested by Kandilakis [17] for cwk ( X )-valued multimeasures anda Banach space. In that case the set in the braces was closed and the additional closure issuperfluous. The symbol ( s ) used here indicates that the integral is constructed using theselections of N . Remark 2.2.
L. DI PIAZZA, K. MUSIA L, A. R. SAMBUCINI • A multimeasure N : Σ → c ( X ) is called rich if N ( A ) = { n ( A ) , n ∈ S N } . By a result ofCost´e, quoted in [15, Theorem 7.9], this is verified for multimeasures which take as theirvalues weakly compact convex subsets of a Banach space X , or cb ( X )-valued when X is aBanach space possessing the RNP ( [11, Th´eor`eme 1]). • If N is a cwk ( X ) valued multimeasure of bounded variation and X is a Banach space, then S N = ∅ (see for example [14, 16]). Moreover, by [17, Theorem 3.2], ( s ) Z E f dN ∈ cwk ( X ). • If f, g are Aumann-BDS-integrable, then( s ) Z E ( f + g ) dN ⊆ ( s ) Z E f dN + ( s ) Z E g dN . The example of g = − f shows that the equality fails in general. In particular, the integralin not additive. • We know from [17, Theorem 3.3] that in case of a multimeasure N with cwk ( X ) values in aBanach space X and bounded f , we have for every x ′ ∈ X ′ and every E ∈ Σs (cid:18) x ′ , ( s ) Z E f dN (cid:19) = Z E f + ds ( x ′ , N ) + Z E f − ds ( − x ′ , N )(2) = Z E f + ds ( x ′ , N ) + Z E f − ds ( x ′ , − N ) in general = Z E f + ds ( x ′ , N ) − Z E f − ds ( x ′ , N )= Z E f ds ( x ′ , N ) . But it follows from that proof that the above equality holds true in each case when S N = ∅ ,that is even when X is a locally convex space and N is not weakly compactly valued.In particular, if f ≥
0, then(3) s (cid:18) x ′ , ( s ) Z E f dN (cid:19) = Z E f ds ( x ′ , N )and the integral is additive for non-negative functions. Moreover, it follows from (2) that theintegral is a multimeasure, for every Aumann-BDS-integrable f .As noticed in [30, Remark 3.2], if f is negative, then the equality (3) may fail. In fact, according to(2), we have then s (cid:18) x ′ , ( s ) Z E ( − dN (cid:19) = Z E ds ( − x ′ , N ) = s ( − x ′ , N ( E )) in general = − s ( x ′ , N ( E )) = Z E − ds ( x ′ , N ) . Now we would like to define an integral with respect to a multimeasure that would be independentof selections of the multimeasure but would be consistent with earlier definitions via selections. Weknow already ( [17, 30]) that if X is a Banach space and a non-negative θ : Ω → R is integrable withrespect to N : Σ → cwk ( X ), then for every E ∈ Σ there exists W E ∈ cwk ( X ) such that for every x ′ ∈ X ′ holds true the equality s ( x ′ , W E ) = Z E θ ds ( x ′ , N ) . We take this property as the definition of the integral of a non-negative function with respect to anarbitrary multimeasure N : Σ → c ( X ). Our approach is close to Lewis’ [18] functional equivalentdefinition of an integral with respect to a vector measure.It is worth to remember that the Lewis definition in [18] was also considered by Kluv´anek in [19] (seealso [27]). We call the integral that we are going to define a multivalued-Bartle-Dunford-Schwartzintegral since Bartle, Dunford and Schwartz were the first who considered such a kind of integrationin the vector case. ADON-NIKOD´YM REPRESENTATIONS FOR THE
BDS m -INTEGRAL 5 Definition 2.3.
Let N : Σ → c ( X ) be a multimeasure. If f : Ω → R is a non-negative measurablefunction, we say that f is multivalued-Bartle-Dunford-Schwartz integrable with respect to N (shortly BDS m - integrable with respect to N ) in c ( X ) ( cb ( X ) , cwk ( X ) , ck ( X )), if for every E ∈ Σ there exists C E ∈ c ( X ) , ( cb ( X ), cwk ( X ) , ck ( X )) such that for every x ′ ∈ X ′ s ( x ′ , C E ) = Z E f ds ( x ′ , N ) . (4)We set C E := Z E f dN .We say that a measurable f : Ω → R is multivalued–Bartle-Dunford-Schwartz integrable with re-spect to N in c ( X ) ( cb ( X ) , cwk ( X ) , ck ( X )), if f + and f − are BDS m -integrable with respect to N in c ( X ) ( cb ( X ) , cwk ( X ) , ck ( X )). Then, for every E ∈ Σ , we define the integral of f with respect to N as Z E f dN := Z E f + dN + Z E f − d ( − N ) . Equivalently, f : Ω → R is BDS m -integrable with respect to N in c ( X ) , ( cb ( X ), cwk ( X ) , ck ( X )), iffor each E ∈ Σ there exists M f ( E ) ∈ c ( X ) ( cb ( X ), cwk ( X ) , ck ( X )) such that for every x ′ ∈ X ′ (5) s ( x ′ , M f ( E )) = Z E f + ds ( x ′ , N ) + Z E f − ds ( x ′ , − N )and the right hand side of (5) makes sense. We write then Z E f dN := M f ( E ).The above definition is consistent with the property described in [17, Theorem 3.3]. Since s ( x ′ , ± N ) : Σ → ( −∞ , + ∞ ] are σ -finite measures for every x ′ then, by (5), the set functions s ( x ′ , M f ) are σ -finitemeasures with values in ( −∞ , + ∞ ]. Consequently, M f is a multimeasure.Let us notice that if N is a vector measure (say N = ν ) then the right hand side of (5) looks as follows: s ( x ′ , M f ( E )) = Z E f d h x ′ , ν i . One can easily check that the integral on the right hand side has to be finite and M is a vector measure(see Lemma 2.7). Remark 2.4. • Assume that S N = ∅ (see for example Remark 2.2). Observe that if f is a bounded, mea-surable, Aumann-BDS-integrable function whose integral belongs to c ( X ) then f is BDS m -integrable with respect to N and( s ) Z E f dN = M f ( E ) . • If f is a BDS m -integrable with respect to N function then M f ( A ∪ B ) = M f ( A ) ⊕ M f ( B )for every A, B ∈ Σ with A ∩ B = ∅ . In fact, for every x ′ ∈ X ′ , we have: s (cid:18) x ′ , Z A ∪ B f dN (cid:19) = Z Ω ( f χ A ∪ B ) + ds ( x ′ , N ) + Z Ω ( f χ A ∪ B ) − ds ( x ′ , − N )= Z Ω ( f + χ A + f + χ B ) ds ( x ′ , N ) + Z Ω ( f − χ A + f − χ B ) ds ( x ′ , − N )= s (cid:18) x ′ , Z A f dN (cid:19) + s (cid:18) x ′ , Z B f dN (cid:19) = s ( x ′ , M f ( A ) ⊕ M f ( B )) . So, for every x ′ ∈ X ′ , it is s ( x ′ , M f ( A ∪ B )) = s ( x ′ , M f ( A )) + s ( x ′ , M f ( B )) = s ( x ′ , M f ( A ) ⊕ M f ( B )). Remark 2.5.
Let X be a Banach space and N : Σ → cb ( X ) be a d H -multimeasure. If f is ameasurable function such that there exists a sequence of simple functions ( f n ) n which pointwise L. DI PIAZZA, K. MUSIA L, A. R. SAMBUCINI converges to f and the sequences ( Z E f ± n dN ) n are Cauchy in ( cb ( X ) , d H ), then f is BDS m integrablewith respect to N .Notice first that if f = P ni =1 a i E i is measurable with non-negative a i , i = 1 , . . . , n , and N is cb ( X )-valued, then Z E f dN = n M i =1 a i N ( E ∩ E i ) for every E ∈ Σ , since for every x ′ ∈ X ′ the support functionis additive with respect to the Minkowski addition.In the general case, if a sequence ( f n ) n of simple functions converges pointwise to f , then ( f ± n ) n converges to f ± . We consider first f + . ( f + n ) n converges poitwise to f + and for every x ′ ∈ X ′ and forevery E ∈ Σ (cid:18)Z E f + n ds ( x ′ , N ) (cid:19) n = (cid:18) s ( x ′ , Z E f + n dN ) (cid:19) n . Since (cid:18)Z E f + n dN (cid:19) n is Cauchy in ( cb ( X ) , d H ), by the completeness of the hyperspace for every E ∈ Σ there exists M + ( E ) ∈ cb ( X ) such thatlim n →∞ d H (cid:18)Z E f + n dN, M + ( E ) (cid:19) = 0 . We apply now [18, Lemma 2.3] to f + n , f + and s ( x ′ , N ) and we obtain that f + is integrable with respectto s ( x ′ , N ) andlim n →∞ Z E f + n ds ( x ′ , N ) = Z E f + ds ( x ′ , N ) , uniformly with respect to E ∈ Σ. Since d H (cid:18)Z E f + n dN, M + ( E ) (cid:19) = sup x ′ ∈ B X ′ (cid:12)(cid:12)(cid:12)(cid:12) s ( x ′ , Z E f + n dN ) − s ( x ′ , M + ( E )) (cid:12)(cid:12)(cid:12)(cid:12) we have that s ( x ′ , M + ( E )) = Z E f + ds ( x ′ , N ) . For the negative part f − we apply the same construction using s ( x ′ , − N ); in this way we obtainanalogously M − ( E ). Finally, considering M + ( E ) ⊕ M − ( E ) we obtain s ( x ′ , M + ( E ) ⊕ M − ( E )) = Z E f + ds ( x ′ , N ) + Z E f − ds ( x ′ , − N ) . Similarly, if f is a BDS m -integrable function with respect to a d H -multimeasure N , then there existsa sequence ( f n ) n of simple functions that is pointwise convergent to f and the sequence (cid:18)Z E f n dN (cid:19) n is Cauchy in ( cb ( X ) , d H ). See Remark 5.3 for the proof. Proposition 2.6. If N is a positive multimeasure, then the BDS m -integral with respect to N is asublinear function of its integrands. ADON-NIKOD´YM REPRESENTATIONS FOR THE
BDS m -INTEGRAL 7 Proof.
Indeed, assume that f, g are
BDS m -integrable with respect to N and a, b ≥
0. Then, s (cid:18) x ′ , Z E ( af + bg ) dN (cid:19) = Z E ( af + bg ) + ds ( x ′ , N ) + Z E ( af + bg ) − ds ( − x ′ , N ) ≤ Z E ( af + + bg + ) ds ( x ′ , N ) + Z E ( af − + bg − ) ds ( − x ′ , N )= a (cid:20)Z E f + ds ( x ′ , N ) + Z E f − ds ( − x ′ , N ) (cid:21) + b (cid:20)Z E g + ds ( x ′ , N ) + Z E g − ds ( − x ′ , N ) (cid:21) = as (cid:18) x ′ , Z E f dN (cid:19) + bs (cid:18) x ′ , Z E g dN (cid:19) . (cid:3) The following lemma is essential for our further investigation:
Lemma 2.7.
Let
M, N : Σ → c ( X ) be two multimeasures possessing control measure µ and such that M ( E ) = Z E θ dN , for every E ∈ Σ . Then M is pointless if and only if N is pointless. Equivalently, M is a vector measure if and only if N is a vector measure.Proof. Assume that N ( E ) = { κ ( E ) } for every E ∈ Σ and κ : Σ → X is a vector measure. Then, wehave for each E ∈ Σ and each x ′ ∈ X ′ s ( x ′ , M ( E )) = Z E θ + ds ( x ′ , N ) + Z E θ − ds ( x ′ , − N ) = Z E θ + d h x ′ , κ i + Z E θ − d h x ′ , − κ i = Z E θ + d h x ′ , κ i − Z E θ − d h x ′ , κ i = Z E θ d h x ′ , κ i . By our assumption the integral Z E θ d h x ′ , κ i exists and has to be finite. If not, then the equality Z E θ d h x ′ , κ i = + ∞ yields s ( − x ′ , M ( E )) = −∞ , what is impossible. Thus, M has only bounded setsas its values. In such a case the expression on the right hand side of (6) is a linear function on X ′ .Hence the same holds true for x ′ −→ s ( x ′ , M ( E )). But that means that M is a vector measure.A similar situation takes place if M ( E ) := { ν ( E ) } , E ∈ Σ , where ν is a vector measure. By theassumption, we have ∀ E ∈ Σ, ∀ x ′ ∈ X ′ , h x ′ , ν ( E ) i = Z E θ + ds ( x ′ , N ) + Z E θ − ds ( x ′ , − N ) . Let A := { ω : θ ( ω ) > } . By the classical Radon-Nikodym theorem there exists a function f x ′ suchthat(6) ∀ E ∈ Σ A , ∀ x ′ ∈ X ′ , s ( x ′ , N ( E )) = Z E f x ′ dµ . If E ∈ Σ A , then ∀ E ∈ Σ A , ∀ x ′ ∈ X ′ , h x ′ , ν ( E ) i = Z E θ + ds ( x ′ , N ) . Since for every E ∈ Σ A , the function x ′ −→ h x ′ , ν ( E ) i is linear, the same holds true for x ′ −→ Z E θ + ds ( x ′ , N ). Consequently, if a, b ∈ R and x ′ , y ′ ∈ X ′ , then Z E θ + f ax ′ + by ′ dµ = Z E θ + ds ( ax ′ + by ′ , N ) == Z E θ + d [ a · s ( x ′ , N ) + b · s ( y ′ , N )] = Z E θ + [ af x ′ + bf y ′ ] dµ . L. DI PIAZZA, K. MUSIA L, A. R. SAMBUCINI As E ∈ Σ A is arbitrary and θ + | A >
0, we obtain the equality f ax ′ + by ′ = af x ′ + bf y ′ µ − a.e. Then x ′ −→ s ( x ′ , N ( E )) is linear by (6) and this proves that N is a vector measure on Σ A . Similarlyfor A c . (cid:3) Proposition 2.8.
Let
M, N : Σ → c ( X ) be two multimeasures and assume that N is pointless. If θ : Ω → R is a measurable function such that for each E ∈ Σ and x ′ ∈ X ′ (7) s ( x ′ , M ( E )) = Z E θ ds ( x ′ , N ) , then θ is non negative N -almost everywhere.Proof. Let θ = θ + − θ − and let H := { ω ∈ Ω : θ ( ω ) − > } ∈ Σ . Then, we obtain for every E ∈ Σ H the equality s ( x ′ , M ( E )) = Z E − θ − ds ( x ′ , N ) . It is enough to prove that N ( H ) = { } . We suppose, by contradiction, that N ( H ) = { } . Then x ′ −→ s ( x ′ , M ( E )) is sublinear for each E ∈ Σ H and also x ′ −→ − s ( x ′ , M ( E )) is sublinear, because − s ( x ′ , M ( E )) = R E θ − ds ( x ′ , N ). Hence x ′ → s ( x ′ , M ( E )) is linear. Therefore0 = s (0 , M ( E )) = s ( x ′ − x ′ , M ( E )) = s ( x ′ , M ( E )) + s ( − x ′ , M ( E ))and so s ( − x ′ , M ( E )) = − s ( x ′ , M ( E )) = ±∞ , what yields the linearity of x ′ −→ s ( x ′ , M ( E )). Butthat is possible only if M ( E ) is one point set. This however forces N to be a vector measure on Σ H ,what contradicts the pointlessness of N . (cid:3) Remark 2.9.
It follows from Proposition 2.8 that an integral defined by the equality (7) does notpresent the proper approach to integrability with respect to a pointless multimeasure, since only non-negative functions could be integrable.If in Proposition 2.8 N restricted to an element F / ∈ N ( N ) is a vector measure, then M restricted to F is a vector measure (see Lemma 2.7) and θ | F is not necessarily non-negative.3. Control measures
Our aim is to determine when a multimeasure M can be seen as an integral of a scalar function withrespect to a given multimeasure N . We obtain our results under the assumption of the existence ofcontrol measures for the considered multimeasures. In the case of d H -multimeasures control measuresalways exist (see [1]). But in the case of an arbitrary multimeasure that is not obvious. Since we wereunable to find that topic in the literature, we present the necessary facts here. The proofs below arein fact copies of the corresponding proofs from [23], where vector measures were under consideration. Definition 3.1.
A multimeasure M : Σ → c ( X ) satisfies the countable chain condition (ccc) if eachfamily of pairwise disjoint not M -null sets is at most countable. Lemma 3.2.
Assume that
M, N : Σ → c ( X ) are two multimeasures such that M ≪ N . If N satisfies(ccc), then for every A ∈ Σ \ N ( M ) there exists B ∈ Σ \ N ( M ) such that B ⊂ A and N ≪ M on Σ B .Proof. Assume that there is a set A ∈ Σ \ N ( M ) such that for every B ∈ Σ A \ N ( M ) there exists D ∈ [ N ( M ) ∩ B ] \ N ( N ). The lemma of Kuratowski-Zorn and (ccc) give the existence of at mostcountable maximal family { D n } n of pairwise disjoint sets D n ∈ [ N ( M ) ∩ A ] \ N ( N ).One can easily check that S n D n ∈ N ( M ). Since A / ∈ N ( M ), we have A \ S n D n / ∈ N ( M ), but thiscontradicts the maximality of { D n } n . (cid:3) ADON-NIKOD´YM REPRESENTATIONS FOR THE
BDS m -INTEGRAL 9 Lemma 3.3. If N : Σ → c ( X ) is a multimeasure satisfying (ccc), then there exists at most countablefamily { x ′ n : n ∈ N } ⊂ B X ′ satisfying the equality (8) \ n N [ s ( x ′ n , N )] = \ x ′ ∈ X ′ N [ s ( x ′ , N )] . Proof.
It follows from Lemma 3.2 that for every x ′ ∈ X ′ there exists D x ′ ∈ Σ \ N [ s ( x ′ , N )] such that N ≪ s ( x ′ , N ) on Σ D x ′ . The lemma of Kuratowski-Zorn and (ccc) guarantee existence of at mostcountable maximal family { D n } n of disjoint sets D n corresponding to measures s ( x ′ n , N ).Let D = S n D n . If A ∈ Σ and A ∩ D = ∅ , then clearly A ∈ N [ s ( x ′ , N )], for every x ′ ∈ B X ′ .Let now A be an s ( x ′ n , N )-null set, for every n . We have A ∩ D n ∈ N ( N ) for all n and so A ∩ D n ∈N [ s ( x ′ , N )] for every x ′ . Consequently, A ∩ D ∈ N [ s ( x ′ , N )] for every x ′ . Hence, A = ( A \ D ) ∪ ( A ∩ D ) ∈ N [ s ( x ′ , N )] . That proves (8). (cid:3)
Theorem 3.4.
A multimeasure N : Σ → c ( X ) has a finite control measure if and only if it satisfies(ccc). Then, there exists a control measure that is equivalent to N .Proof. It is obvious that the existence of a control measure yields (ccc) of N . So assume that N satisfies (ccc). For each x ′ ∈ X ′ let ν x ′ : Σ → [0 , + ∞ ) be a measure equivalent to s ( x ′ , N ). ByLemma 3.3nthere exist x ′ n ∈ X ′ , n ∈ N , such that N ( N ) = \ x ′ ∈ X ′ N [ s ( x ′ , N )] = \ n N [ s ( x ′ n , N )] = \ n N ( ν x ′ n ) . The measure µ ( E ) := ∞ X n =1 n ν x ′ n ( E )1 + ν x ′ n ( Ω )(9)is the required control measure for N that is equivalent to N . (cid:3) Question 3.5.
Let X be a Banach space. Does every multimeasure M : Σ → cb ( X ) have a controlmeasure? In case of a locally convex X even vector measures may not admit any control measure. Theorem 3.6. If X ′ is weak ′ -separable, then every multimeasure M : Σ → c ( X ) satisfies (ccc).Proof. Let { x ′ n : n ∈ N } be weak ′ -dense in X ′ . Suppose there exists a multimeasure M : Σ → c ( X )and an uncountable family { B α ∈ Σ : α ∈ A } of pairwise disjoint sets with M ( B α ) = { } .Without loss of generality we may assume that the family is ordered by the ordinals less than ω . Dueto the countability of the weak ′ -dense set, there exists β < ω such that s ( x ′ n , M ( B α )) = 0, for every α > β and every n ∈ N . Hence, if α > β and x ∈ B α , then h x ′ n , x i ≤ n . It follows that h x ′ , x i ≤ x ′ ∈ X ′ , if x ∈ B α and α > β . This is of course impossible for x = 0. (cid:3) Arbitrary multimeasures
Now we start to examine the problem when a multimeasure M can be represented as an integral of ascalar function with respect to a given multimeasure N .In the case of vector measures with values in a locally convex space a Radon-Nikod´ym theorem forthe Bartle-Dunford-Schwartz integral was obtained by Musia l in [22]. If Y is a locally convex spaceand ν, κ : Σ → Y are vector measures, then the following definitions were formulated in [22]: usac): ν is uniformly scalarly absolutely continuous (usac) with respect to κ , if for each ε > δ > y ′ ∈ Y ′ and each E ∈ Σ , the inequality | y ′ κ | ( E ) < δ yields | y ′ ν | ( E ) < ε . We denote it by ν ≪ κ . usd): ν is uniformly scalarly dominated (usd) by κ , if there exists b ∈ R + such that ∀ y ′ ∈ Y ′ , ∀ E ∈ Σ it is | y ′ ν | ( E ) ≤ b | y ′ κ | ( E ) . sub): ν is subordinated to κ , if there exists d ∈ R + such that for every E ∈ Σν ( E ) ∈ d aco R ( κ E ) . We say that a vector measure ν : Σ → Y has locally a property with respect to a vector measure κ : Σ → Y , if for each E ∈ Σ \ N ( κ ) there exists F ⊂ E with F ∈ Σ \ N ( κ ) such that ν has thisproperty with respect to κ on the set F .In order to find proper conditions guarateeing the differentiation of an arbitrary cb ( X )-valued multi-measure M with respect to N , we must adapt properly the definitions given for vector measures. Definition 4.1.
Given two multimeasures
M, N : Σ → cb ( X ) we say that: (usac): M is uniformly scalarly absolutely continuous with respect to N ( usac or M ≪ N ),if there exists A ∈ Σ such that ∀ ε > , ∃ δ > ∀ α, β ∈ R , ∀ x ′ , y ′ ∈ B X ′ , ∀ E ∈ Σ (10) [ | αs ( x ′ , N ) + βs ( y ′ , N ) | ( E ∩ A ) + | αs ( x ′ , − N ) + βs ( y ′ , − N ) | ( E ∩ A c ) ≤ δ ] ⇒⇒ | αs ( x ′ , M ) + βs ( y ′ , M ) | ( E ) ≤ ε ; (usd): M is uniformly scalarly dominated ( usd ) by a multimeasure N , if there exist c ∈ R and A ∈ Σ such that ∀ α, β ∈ R , ∀ x ′ , y ′ ∈ B X ′ , ∀ E ∈ Σ | αs ( x ′ , M ) + βs ( y ′ , M ) | ( E ) ≤≤ c | αs ( x ′ , N ) + βs ( y ′ , N ) | ( E ∩ A ) + c | αs ( x ′ , − N ) + βs ( y ′ , − N ) | ( E ∩ A c ) ; (uss): M is uniformly scalarly subordinated ( uss ) to N , if there exist d ∈ R + and A ∈ Σ suchthat ∀ α, β ∈ R , ∀ x ′ , y ′ ∈ B X ′ , ∀ E ∈ Σαs ( x ′ , M ( E )) + βs ( y ′ , M ( E )) ∈∈ d aco { [ αs ( x ′ , N ( F ∩ A )) + βs ( y ′ , N ( F ∩ A ))] : F ∈ Σ E } ++ d aco { [ αs ( x ′ , − N ( F ∩ A c )) + βs ( y ′ , − N ( F ∩ A c ))] : F ∈ Σ E } . Remark 4.2. If M and N are vector measures then the uniform scalar absolute continuity andthe uniform scalar domination of M with respect to N coincide respectively with the correspondingdefinitions given for vector measures, therefore we use the same notation for the two notions. In caseof uss the definition looks different with respect to sub and so we decided to change the name.They actually are equivalent if we assume that X is a quasi-complete locally convex space. This isdue to the fact that for every E ∈ Σ and for every x ′ ∈ X ′ aco {h x ′ , R ( κ E ) i} = h x ′ , aco ( R ( κ E )) i with(11) h x ′ , R ( κ E ) i := {h x ′ , w i : w ∈ R ( κ E ) } . Here the inclusion ⊃ is a consequence of the continuity of x ′ ∈ X ′ : h x ′ , aco ( R ( κ E )) i ⊂ h x ′ , aco ( R ( κ E )) i = aco {h x ′ , R ( κ E ) i} . While for the reverse inclusion, we can observe that the set aco R ( κ E ) is symmetric and weaklycompact, due to [20, Theorem IV.6.1]. So we have h x ′ , aco R ( κ E ) i = [ − a, a ]. The inclusion R ( κ E ) ⊂ aco R ( κ E ) yields h x ′ , R ( κ E ) i ⊂ [ − α, α ] and so aco h x ′ , R ( κ E ) i ⊂ [ − α, α ]. Proposition 4.3.
For arbitrary multimeasures
M, N : Σ → cb ( X ) the properties usac and usd areequivalent and uss implies each of them.Proof. usd) ⇒ usac) is obvious. (usac) ⇒ (usd) Let A ∈ Σ , ε > δ > | αs ( x ′ , N ) + βs ( y ′ , N ) | ( E ) < δ implies | αs ( x ′ , M ) + βs ( y ′ , M ) | ( E ) < ε .If | αs ( x ′ , N )+ βs ( y ′ , N ) | ( E ) = 0, then M ≪ N yields | αs ( x ′ , M )+ βs ( y ′ , M ) | ( E ) < ε for every ε > . ADON-NIKOD´YM REPRESENTATIONS FOR THE
BDS m -INTEGRAL 11 So usd follows.Suppose now that | αs ( x ′ , N ) + βs ( y ′ , N ) | ( E ) >
0. Letˆ x := δx ′ [2 | αs ( x ′ , N ) + βs ( y ′ , N ) | ( E )] − ˆ y := δy ′ [2 | αs ( x ′ , N ) + βs ( y ′ , N ) | ( E )] − Then | αs (ˆ x, N ) + βs (ˆ y, N ) | ( E ) < δ and consequently | αs (ˆ x, M ) + βs (ˆ y, M ) | ( E ) ≤ ε. It follows that,if we take c = 2 ε/δ , then | αs ( x ′ , M ) + βs ( y ′ , M ) | ( E ) ≤ c | αs ( x ′ , N ) + βs ( y ′ , N ) | ( E ) . In a similar way one obtains the required inequalities for very E ∈ Σ A c . That proves usd . (uss) ⇒ (usd) By definition, for every F ∈ Σ E , we have αs ( x ′ , M ( E )) + βs ( y ′ , M ( E )) ≤≤ sup d aco { [ αs ( x ′ , N ( F ∩ A )) + βs ( y ′ , N ( F ∩ A ))] } ++ sup d aco { [ αs ( x ′ , − N ( F ∩ A c )) + βs ( y ′ , − N ( F ∩ A c ))] } ≤≤ d | αs ( x ′ , N ) + βs ( y ′ , N ) | ( E ∩ A ) ++ d | αs ( x ′ , − N ) + βs ( y ′ , − N ) | ( E ∩ A c ) . So the assertion follows. (cid:3)
Theorem 4.4.
Let
M, N : Σ → cb ( X ) be two consistent multimeasures possessing control measures.Then the following are equivalent: (RN b ): There exists a bounded measurable function θ : Ω → R such that for every E ∈ Σ wehave (12) M ( E ) = Z E θ dN ; (4.4.i): M is uniformly scalarly dominated by N ; (4.4.ii): M is uniformly scalarly absolutely continuous with respect to N ; (4.4.iii): M is uniformly scalarly subordinated to N .Proof. Let µ be a finite control measure for N . Without loss of generality, we may assume that µ isalso a control measure for M . Let H ∈ Σ be a set as in the definition of the consistency for M and N . We divide the proof into the pointless and vector parts. (Pointless part): We assume for simplicity that H = Ω . (4.4.i) ⇒ (RN b ): By the classical Radon-Nikod´ym theorem, for every x ′ ∈ B X ′ , thereexist two measurable real functions f x ′ and g x ′ such that, ∀ E ∈ Σ, (13) s ( x ′ , M ( E )) = Z E f x ′ dµ and s ( x ′ , N ( E )) = Z E g x ′ dµ . • Let A be the set satisfying the definition of uniform scalar domination of M withrespect to N . If E ∈ Σ A and α, β ∈ R , then Z E | αf x ′ + βf y ′ | dµ = | αs ( x ′ , M ) + βs ( y ′ , M ) | ( E ) ≤ c | αs ( x ′ , N ) + βs ( y ′ , N ) | ( E )= c Z E | αg x ′ + βg y ′ | dµ. So | αf x ′ + βf y ′ | ≤ c | αg x ′ + βg y ′ | µ − a.e. on A. Hence (see also [22, Lemma]), f x ′ cg x ′ = f y ′ cg y ′ µ − a.e. on the set { ω ∈ Ω : g x ′ ( ω ) g y ′ ( ω ) = 0 } ∩ A .
According to [31, Theorem 7.35.2] (see also [22, Lemma] for a short proof) thereexists a measurable θ : A → [ − ,
1] such that f x ′ = c θ g x ′ µ -a.e. on the set { ω ∈ A : g x ′ ( ω ) g y ′ ( ω ) = 0 } for each x ′ ∈ X ′ separately. The equality on the set { ω ∈ A : g x ′ ( ω ) = 0 } is obvious (notice that | f x ′ | ≤ c | g x ′ | a.e. on A ). So it followsthat bs ( x ′ , M ( E )) = Z E c θ ds ( x ′ , N )for each E ∈ Σ A and x ′ ∈ X ′ . We can prove that θ ≥ N -a.e. on A , thanks toProposition 2.8. • Let now F ∈ Σ A c . For every α, β ∈ R and for every x ′ , y ′ ∈ X ′ we have Z F | αf x ′ + βf y ′ | dµ = | αs ( x ′ , M ) + βs ( y ′ , M ) | ( F ) ≤ c | αs ( x ′ , − N ) + βs ( y ′ , − N ) | ( F )= c Z F | αg − x ′ + βg − y ′ | dµ. A similar calculation gives the equality f x ′ cg − x ′ = f y ′ cg − y ′ µ − a.e. on the set { ω ∈ A c : g − x ′ ( ω ) g − y ′ ( ω ) = 0 } and then s ( x ′ , M ( F )) = Z F c θ ds ( x ′ , − N ) . As before, θ : A c → R is non-negative N -a.e.. Thus, set θ = cθ − cθ , for every E ∈ Σs ( x ′ , M ( E )) = s ( x ′ , M ( E ∩ A )) + s ( x ′ , M ( E ∩ A c )) == Z E θ + ds ( x ′ , N ) + Z E θ − ds ( x ′ , − N ) , what means that θ is a Radon-Nikod´ym derivative of M with respect to N . ( RN b ) ⇒ (4.4.iii): We set A := { ω ∈ Ω : θ ( ω ) ≥ } , and d > sup ω ∈ Ω | θ ( ω ) | . Since θ is BDS m -integrable with respect to N , by Definition 2.3 we have for each E ∈ Σ theequality αs ( x ′ , M ( E ∩ A )) + βs ( y ′ , M ( E ∩ A )) = Z E ∩ A θ + d [ αs ( x ′ , N ) + βs ( y ′ , N )] . Let x ′ , y ′ , α, β be fixed and let B, C generate the Hahn decomposition of αs ( x ′ , N ) + βs ( y ′ , N ). Then, αs ( x ′ , M ( E ∩ A ∩ B )) + βs ( y ′ , M ( E ∩ A ∩ B )) ∈ co [ θ + ( E ∩ A ∩ B )] · [ αs ( x ′ , N ( E ∩ A ∩ B )) + βs ( y ′ , N ( E ∩ A ∩ B ))]Similarly, αs ( x ′ , M ( E ∩ A ∩ C )) + βs ( y ′ , M ( E ∩ A ∩ C )) ∈ co [ θ + ( E ∩ A ∩ C )] · [ αs ( x ′ , N ( E ∩ A ∩ C )) + βs ( y ′ , N ( E ∩ A ∩ C ))]All together yields αs ( x ′ , M ( E ∩ A )) + βs ( y ′ , M ( E ∩ A )) ∈ co [ θ + ( E ∩ A )] · (cid:16) [ αs ( x ′ , N ( E ∩ A ∩ B )) + βs ( y ′ , N ( E ∩ A ∩ B ))]+ [ αs ( x ′ , N ( E ∩ A ∩ C )) + βs ( y ′ , N ( E ∩ A ∩ C ))] (cid:17) = co [ θ + ( E ∩ A )] · [ αs ( x ′ , N ( E ∩ A )) + βs ( y ′ , N ( E ∩ A ))] ⊂ d aco { αs ( x ′ , N ( F ∩ A )) + βs ( y ′ , N ( F ∩ A )) : F ∈ Σ E } . ADON-NIKOD´YM REPRESENTATIONS FOR THE
BDS m -INTEGRAL 13 In the same way one obtains the inclusion αs ( x ′ , M ( E ∩ A c )) + βs ( y ′ , M ( E ∩ A c )) ⊂ d aco { αs ( x ′ , − N ( F ∩ A c )) + βs ( y ′ , − N ( F ∩ A c )) : F ∈ Σ E } . The remaing equivalences follow from Proposition 4.3. (Vector part):
We assume now that H c = Ω . In virtue of Remark 4.2, the equivalence ofthe four conditions is a consequence of [22, Theorem 1]. (cid:3) Similarly to the vector case we say that a multimeasure M : Σ → cb ( X ) has locally a propertywith respect to a multimeasure N : Σ → cb ( X ), if for each E ∈ Σ \ N ( N ) there exists F ⊂ E with F ∈ Σ \ N ( N ) such that M has the property with respect to N on the set F .Using the local properties we obtain: Theorem 4.5.
Let
M, N : Σ → c ( X ) be two σ -bounded consistent multimeasures possessing controlmeasures. Then the following are equivalent: (RN): There exists a measurable function θ : Ω → R such that ∀ E ∈ Σ M ( E ) = Z E θ dN ; (4.5.i): M is locally uniformly scalarly dominated by N ; (4.5.ii): M is locally uniformly scalarly absolutely continuous with respect to N ; (4.5.iii): M is locally uniformly scalarly subordinated to N .Proof. Assume that Ω = S n Ω n ∪ B , where Ω n -s are pairwise disjoint, B ∈ N ( µ ) and the restrictionof M and N to each Ω n is cb ( X )-valued. First we assume that M, N are cb ( X )-valued on Σ .Analogously to Theorem 4.4 we have to divide the proof into the pointless and vector parts. Thevector part follows immediately from [22, Theorem 2], so we prove here only the pointless part. • The equivalences (4.5.i) ⇔ (4.5.ii) ⇔ (4.5.iii) follow from the corresponding equivalencesin Theorem 4.4. • Also (RN) ⇒ (4.5.i) is obvious (it is enough to take for each E ∈ Σ \ N ( N ) a set F ⊂ E such that F ∈ Σ \ N ( N ) and θ is bounded on F ). • Now we are going to prove that (4.5.i) ⇒ (RN) .Let µ be a control measure for N . By definition there exist E ∈ Σ \ N ( µ ) such that M isscalarly dominated on E by N . We denote by H := { E ∈ Σ \ N ( µ ) : ∃ A E ∈ Σ E ∀ F ∈ Σ E , ∀ x ′ , y ′ ∈ X ′ , ∀ α, β ∈ R , | αs ( x ′ , M ) + βs ( y ′ , M ) | ( F ) ≤ | αs ( x ′ , N ) + βs ( y ′ , N ) | ( F ∩ A E ) + | αs ( x ′ , − N ) + βs ( y ′ , − N ) | ( F \ A E ) } . By the completeness of the algebra Σ/ N ( µ ) there is E ∈ Σ such that its equivalence classis the least upper bound of H in Σ/ N ( µ ). If H is empty we choose E = ∅ .Then we consider the class H of all sets E ∈ Σ Ω \ E \ N ( µ ) in which we have the scalarlydomination with respect to c = 2 and we choose analogously E .After n steps we have already sets in this way E , . . . , E n and A E ⊂ E , . . . , A E n ⊂ E n suchthat for each k ≤ n the inequality | αs ( x ′ , M ) + βs ( y ′ , M ) | ( F ) ≤ k {| αs ( x ′ , N ) + βs ( y ′ , N ) | ( F ∩ A E k ) + | αs ( y ′ , − N ) + βs ( x ′ , − N ) | ( F \ A E k ) } holds true for all F ∈ Σ E k , for all α, β ∈ R and for all x ′ , y ′ ∈ X ′ . Then we construct E n +1 such that its equivalence class is the least upper bound of H n +1 := { E ∈ Σ Ω \ S nk =1 E k \ N ( λ ) : ∃ A E ∈ Σ E ∀ F ∈ Σ E , ∀ x ′ , y ′ ∈ X ′ , ∀ α, β ∈ R | αs ( x ′ , M ) + βs ( y ′ , M ) | ( F ) ≤ ( n + 1) | αs ( x ′ , N ) + βs ( y ′ , N ) | ( F ∩ A E )+ ( n + 1) | αs ( x ′ , − N ) + βs ( y ′ , − N ) | ( F \ A E ) } . in Σ/ N ( µ ). A few first sets E k may be of measure zero but then we meet the first set E k of positive measure. In this way we obtain a sequence (possibly finite) ( E n ) of sets with E n ∈ H n .Without loss of generality we may assume that the sets cover all Ω . Now we apply Theorem4.4 to each set E n and we get a bounded measurable function θ n such that(14) ∀ E ∈ Σ E n , ∀ x ′ ∈ X ′ , s ( x ′ , M ( E )) = Z E θ + n ds ( x ′ , N ) + Z E θ − n ds ( x ′ , − N ) . Let θ = P ∞ n =1 θ n χ E n be a formal serie since ( E n ) n are pairwise disjoint and let x ′ ∈ X ′ befixed. Then, θ + = P ∞ n =1 θ + n χ E n and θ − = P ∞ n =1 θ − n χ E n . By (14) we have for every E ∈ Σs ( x ′ , M ( E )) = ∞ X n =1 s ( x ′ , M ( E ∩ E n )) == ∞ X n =1 (cid:18)Z E ∩ E n θ + n ds ( x ′ , N ) + Z E ∩ E n θ − n ds ( x ′ , − N ) (cid:19) (15) Let Ω + := { ω ∈ Ω : θ ( ω ) ≥ } , Ω − := Ω \ Ω + . Let P ′ , P ′ , Q ′ , Q ′ ∈ Σ be two Hahndecomposition for s ( x ′ , N ) , s ( x ′ , − N ) respectively. We have s ( x ′ , M ( E ∩ P ′ ∩ Ω + )) = ∞ X n =1 s ( x ′ , M ( E ∩ E n ∩ P ′ ∩ Ω + )) == ∞ X n =1 Z E ∩ E n ∩ P ′ ∩ Ω + θ + n ds ( x ′ , N ) s ( x ′ , M ( E ∩ P ′ ∩ Ω + )) = ∞ X n =1 s ( x ′ , M ( E ∩ E n ∩ P ′ ∩ Ω + )) == ∞ X n =1 Z E ∩ E n ∩ P ′ ∩ Ω + θ + n ds ( x ′ , N ) s ( x ′ , M ( E ∩ Q ′ ∩ Ω − )) = ∞ X n =1 s ( x ′ , M ( E ∩ E n ∩ Q ′ ∩ Ω − )) == ∞ X n =1 Z E ∩ E n ∩ P ′ ∩ Ω − θ − n ds ( x ′ , − N ) s ( x ′ , M ( E ∩ Q ′ ∩ Ω − )) = ∞ X n =1 s ( x ′ , M ( E ∩ E n ∩ Q ′ ∩ Ω − )) == ∞ X n =1 Z E ∩ E n ∩ P ′ ∩ Ω − θ − n ds ( x ′ , − N ) . Each of the series closing the above equalities is convergent with all its terms of the samesign. Hence, they are absolutely convergent. It follows that for all E ∈ Σs ( x ′ , M ( E )) = ∞ X n =1 s ( x ′ , M ( E ∩ E n )) = Z E ∩ P ′ θ + ds ( x ′ , N ) + Z E ∩ P ′ θ + ds ( x ′ , N ) ++ Z E ∩ Q ′ θ − ds ( x ′ , − N ) + Z E ∩ Q ′ θ − ds ( x ′ , − N ) == Z E θ + ds ( x ′ , N ) + Z E θ − ds ( x ′ , − N ) . ADON-NIKOD´YM REPRESENTATIONS FOR THE
BDS m -INTEGRAL 15 Let us consider now the general case. We know already that for each n ∈ N there exists a measurablefunction ξ n : Ω n → R such that(16) ∀ E ∈ Σ Ω n ∀ x ′ ∈ X ′ s ( x ′ , M ( E )) = Z E ξ + n ds ( x ′ , N ) + Z E ξ − n ds ( x ′ , − N ) . Then, we follow the proof presented after formula (14). We have to remember only that we havealways s ( x ′ , M ( E )) > −∞ and so each series appearing in the proof is either divergent to + ∞ orconvergent. (cid:3) Remark 4.6.
It follows from Lemma 2.7 that some introductory assumptions in Theorem 4.4 con-cerning M and N are necessary. If M is pointless but N is not, then M cannot be represented by a BDS m -integral with respect to N . To construct an example let X be a quasi-complete locally convexspace and ν : Σ → X be a non-atomic vector measure.If M is defined by the formula M ( E ) := co R ( ν E ) (or M ( E ) := aco R ( ν E )), then aco R ( ν E ) is aweakly compact set (see [20, Theorem IV.6.1]) and the conditions uss , usd and usac are fulfilled by M and N . Indeed the formula (11) in Remark 4.2, shows that αs ( x ′ , M ( E )) + βs ( x ′ , M ( E )) ∈ d aco {h x ′ , κ ( F ) i : F ∈ R ( κ E ) } , whenever E ∈ Σ and x ′ ∈ X ′ are arbitrary. It remains to prove that M ( E ) := co R ( ν E )is a multimeasure. M is clearly finitely additive: if A, B are disjoint, then M ( A ∪ B ) = co R ( ν A ∪ B ) = co ( R ( ν A ) + R ( ν B )) = co R ( ν A ) ⊕ co R ( ν B ) = M ( A ) ⊕ M ( B ) . It follows that for every x ′ ∈ X ′ , s ( x ′ , M ) : Σ → R is finitely additive. [3, Proposition 3.8] yields thecountable additivity of each s ( x ′ , M ) and so M is a multimeasure. Remark 4.7.
A particular example of a quasi-complete locally convex space is a conjugate Banachspace endowed with the weak ∗ topology σ ( X ′ , X ). Let cw ∗ k ( X ′ ) denote the family of all non-emptyweak ∗ -compact and convex subsets of X ′ . M : Σ → cw ∗ k ( X ′ ) is called a weak ∗ -multimeasure if s ( x, M ( · )) is a measure, for every x ∈ X . It is proved in [26, Theorem 3.4] that X does not con-tain any isomorphic copy of ℓ ∞ if and only if each weak ∗ -multimeasure M : Σ → cw ∗ k ( X ′ ) is a d H -multimeasure. The Radon-Nikod´ym theorem for two weak ∗ -multimeasures formulates exactly asTheorem 4.5, one should only remember that now the conjugate to ( X ′ , σ ( X ′ , X )) is the space X itself. 5. d H -multimeasures and their R˚adstr¨om embeddings Throughout this section we assume that X is a Banach space and we consider now the case of d H -multimeasures. Let us recall that if a Banach space X does not contain any isomorphic copy of c ,then each cb ( X )-valued multimeasure is a d H -multimeasure (see [7]). Examples of cb ( c )-valued mea-sures that are not d H -measures can be found in [25, Example 3.6, Example 3.8].Let now M, N : Σ → cb ( X ) be two d H -multimeasures. We remember that k s ( · , M ( E )) k ∞ =sup x ′ ∈ B X ′ | s ( x ′ , M ( E )) | and, according to (1), we set Y := ℓ ∞ ( B X ′ ). By [22, Theorem 1], for ν := j ◦ M, κ = j ◦ N we have Theorem 5.1. If M, N : Σ → cb ( X ) are two d H -multimeasures, then the following are equivalent: RN j ): There exists a bounded measurable function (measurable function) θ : Ω → R such thatfor all E ∈ Σ and y ′ ∈ ℓ ′∞ ( B X ′ ) h y ′ , j ◦ M ( E ) i = Z E θ d h y ′ , j ◦ N i ; (5.1.i): j ◦ M is (locally) uniformly scalarly absolutely continuous with respect to j ◦ N ; (5.1.ii): j ◦ M is (locally) uniformly scalarly dominated by j ◦ N ; (5.1.iii): j ◦ M is (locally) subordinated to j ◦ N . We observe that even if M and N are d H -multimeasures, the differentiation of M with respect to N isin general not equivalent to differentiation of j ◦ M with respect to j ◦ N . If θ is the Radon-Nikod´ymderivative of M with respect to N , the representation j ◦ M ( E ) = ( BDS ) Z E θ d ( j ◦ N ) for every E ∈ Σ may fail (see the subsequent Example 6.4). In fact one has to take into account also integration withrespect to the measure j ◦ ( − N ). If N is a vector measure, then j ◦ ( − N ) = − j ◦ N and formally themeasure j ◦ ( − N ) is absent in the calculations and the integral with respect to N coincides with itsBDS-integral.In the case of multimeasures that are not vector measures, we have in general j ◦ ( − N ) = − j ◦ N and the presence of the measure j ◦ ( − N ) becomes visible. Next theorem shows the shape of j ◦ N provided M has the Radon-Nikod´ym derivative (in the sense of the integral investigated in this paper)with respect to N . Theorem 5.2.
Let
M, N : Σ → cb ( X ) be two d H -multimeasures and θ : Ω → R be a measurablefunction. Then M ( E ) = Z E θ dN for every E ∈ Σ if and only if for all E ∈ Σ and for all y ′ ∈ ℓ ′∞ ( B X ′ )(17) h y ′ , j ◦ M ( E ) i = Z E θ + d h y ′ , j ◦ N i + Z E θ − d h y ′ , j ◦ ( − N ) i . Equivalently, j ◦ M ( E ) = ( BDS ) Z E θ + d ( j ◦ N ) + ( BDS ) Z E θ − d ( j ◦ ( − N )) . Proof. ⇒ : (5.2.A): We assume first that θ is bounded. Let M ( E ) = Z E θ dN for every E ∈ Σ . According to (5) we have then h e x ′ , j ◦ M ( E ) i = s ( x ′ , M ( E )) = Z E θ + ds ( x ′ , N ) + Z E θ − ds ( − x ′ , N )= Z E θ + d h e x ′ , j ◦ N i + Z E θ − d h e − x ′ , j ◦ N i == Z E θ + d h e x ′ , j ◦ N i + Z E θ − d h e x ′ , j ◦ ( − N ) i . Since θ is bounded, it is j ◦ N integrable (as a BDS-integral) and so there are measures κ , κ : Σ → ℓ ∞ ( B X ′ ) such that h y ′ , κ ( A ) i = Z E θ + d h y ′ , j ◦ N i whenever y ′ ∈ ℓ ′∞ ( B X ′ )and h y ′ , κ ( E ) i = Z E θ − d h y ′ , j ◦ ( − N ) i whenever y ′ ∈ ℓ ′∞ ( B X ′ )Hence, we obtain the equality h e x ′ , j ◦ M ( E ) i = h e x ′ , κ ( E ) i + h e x ′ , κ ( E ) i . But { e x ′ : x ′ ∈ B X ′ } is norming and the measures j ◦ M, κ and κ have weakly relativelycompact ranges. It follows that ∀ y ′ ∈ ℓ ′∞ ( B X ′ ) , ∀ E ∈ Σ h y ′ , j ◦ M ( E ) i = Z E θ + d h y ′ , j ◦ N i + Z E θ − d h y ′ , j ◦ ( − N ) i . (18) ADON-NIKOD´YM REPRESENTATIONS FOR THE
BDS m -INTEGRAL 17 Assume now that the equality (18) is fulfilled. Then, ∀ x ′ ∈ X ′ , ∀ E ∈ Σs ( x ′ , M ( E )) = h e x ′ , j ◦ M ( E ) i = Z E θ + d h e x ′ , j ◦ N i + Z E θ − d h e x ′ , j ◦ ( − N ) i = Z E θ + ds ( x ′ , N ) + Z E θ − ds ( x ′ , − N ) = s ( x ′ , Z E θ dN ) . (5.2.B): Suppose now that θ ≥ M ( E ) = Z E θ dN for every E ∈ Σ .That means that for every x ′ ∈ X ′ and E ∈ Σ we have s ( x ′ , M ( E )) = Z E θ ds ( x ′ , N ) .θ can be represented as θ = P n θχ E n , where the sets E n are pairwise disjoint and each θχ E n is bounded. According to (5.2.A) part, we have for all y ′ ∈ ℓ ′∞ ( B X ′ ) h y ′ , j ◦ M ( E ∩ E n ) i = Z E ∩ E n θ d h y ′ , j ◦ N i , n ∈ N . Since for each y ′ ∈ ℓ ′∞ ( B X ′ ) the set function h y ′ , j ◦ M i is a measure, we obtain theequality P n h y ′ , j ◦ M ( E ∩ E n ) i = h y ′ , j ◦ M ( E ) i . Since for each y ′ ∈ ℓ ′∞ ( B X ′ ) the setfunction h y ′ , j ◦ N i is a measure, we have also X n Z E ∩ E n θ d h y ′ , j ◦ N i = Z E θ d h y ′ , j ◦ N i . (5.2.C): θ is arbitrary and M ( E ) = Z E θ dN for every E ∈ Σ . The proof is obvious. ⇐ : In (17) one should substitute e x ′ instead of y ′ . (cid:3) Remark 5.3.
Let f be a BDS m -integrable function and assume that N is cb ( X )-valued d H -multimeasure.Then there exists a sequence ( f n ) n of simple functions that is pointwise convergent to f and the se-quence (cid:18)Z E f n dN (cid:19) n is Cauchy in ( cb ( X ) , d H ).Indeed, let M ( E ) = R E f dN , E ∈ Σ . According to Theorem 5.2 f + and f − are j ◦ N integrable as(BDS) integrals) and we have for all E ∈ Σ , with E ⊂ supp f + j ◦ M ( E ) = ( BDS ) Z E f + d ( j ◦ N ) . In virtue of [1, Definition 2.5] there exists a sequence of pointwise convergent to f + simple functions h n : supp f + → [0 , ∞ ) such that the sequence (cid:18) ( BDS ) Z E h n d ( j ◦ N ) (cid:19) n is Cauchy in ℓ ∞ ( B X ′ ). Itfollows from Theorem 5.2 that the sequence (cid:18)Z E h n dN (cid:19) n is Cauchy in d H . We repeat the procedurewith f − obtaining a sequence ( g n ) n of simple functions. Setting f n := h n − g n we find the requiredsequence.By Theorem 5.2 and Proposition 2.8 we get Corollary 5.4.
Let
M, N : Σ → cb ( X ) be two consistent d H -multimeasures. Moreover let θ : Ω → R be a measurable function. • If j ◦ M ( E ) = ( BDS ) Z E θ d ( j ◦ N ) for every E ∈ Σ , then θ ≥ N -a.e. and for all E ∈ Σ and all x ′ ∈ X ′ s ( x ′ , M ( E )) = Z E θ ds ( x ′ , N ) . • And conversely, if θ is non-negative and s ( x ′ , M ( E )) = Z E θ ds ( x ′ , N ) for every E ∈ Σ , then j ◦ M ( E ) = ( BDS ) Z E θ d ( j ◦ N ) for every E ∈ Σ . It is our aim to obtain a Radon-Nikod´ym theorem for R˚adstr¨om embeddings of multimeasures interms of the multimeasures themselves, not their R˚adstr¨om embeddings. To achieve it we modify thedefinitions of usac , usd and uss in the following way: Definition 5.5.
Given two multimeasures
M, N : Σ → cb ( X ) we say that: (s-usac): A multimeasure M : Σ → cb ( X ) is strongly uniformly scalarly absolutely continuous ( s - usac ) with respect to a multimeasure N : Σ → cb ( X ), if for each ε > δ > P nm =1 a m x ′ m ∈ span B X ′ and each E ∈ Σ , if | P nm =1 a m s ( x ′ m , N ) | ( E ) ≤ δ ,then (cid:12)(cid:12)(cid:12)(cid:12) n X m =1 a m s ( x ′ m , M ) (cid:12)(cid:12)(cid:12)(cid:12) ( E ) ≤ ε. We denote it by M ≪ s N . (s-usd): A multimeasure M : Σ → cb ( X ) is strongly uniformly scalarly dominated ( s - usd ) bya multimeasure N : Σ → cb ( X ), if there exists a positive d ∈ R such that for every E ∈ Σ and every P nm =1 a m x ′ m ∈ span B X ′ , one has (cid:12)(cid:12)(cid:12)(cid:12) n X m =1 a m s ( x ′ m , M ) (cid:12)(cid:12)(cid:12)(cid:12) ( E ) ≤ d (cid:12)(cid:12)(cid:12)(cid:12) n X m =1 a m s ( x ′ m , N ) (cid:12)(cid:12)(cid:12)(cid:12) ( E ) . (s-uss): A multimeasure M is strongly uniformly scalarly subordinated to N ( s - uss ), if thereexists a positive d ∈ R such that for every E ∈ Σ and every P nm =1 a m x ′ m ∈ span B X ′ , onehas n X m =1 a m s ( x ′ m , M ( E )) ∈ d aco (cid:26) n X m =1 a m s ( x ′ m , N ( F )) : F ∈ Σ E (cid:27) . (19)Using these strong versions of uniform scalar absolute continuity, uniform scalar domination anduniform scalar subordination we are able to prove the following result: Theorem 5.6.
Let
M, N : Σ → cb ( X ) be two consistent d H -multimeasures. Then the following areequivalent (RN j ): There exists a bounded measurable function (measurable function) θ : Ω → R such thatfor all E ∈ Σ and y ′ ∈ ℓ ′∞ ( B X ′ ) h y ′ , j ◦ M ( E ) i = Z E θ d h y ′ , j ◦ N i ; (5.6.j): M is (locally) strongly uniformly scalarly dominated by N ; (5.6.jj): M is (locally) strongly uniformly scalarly absolutely continuous with respect to N ; (5.6.jjj): M is (locally) strongly uniformly scalarly subordinated to N .Proof. (RN j ) ⇒ (5.6.jjj): By Theorem 5.1 the condition (RN j ) implies that j ◦ M is s - uss to j ◦ N . Assume that d > j ◦ M ( E ) ∈ d aco { j ◦ N ( F ) : F ∈ Σ E } ∈ cb ( ℓ ∞ ( B X ′ )) for every E ∈ Σ .
That means that for every y ′ ∈ ℓ ′∞ ( B X ′ ), we have h y ′ , j ◦ M ( E ) i ≤ d sup {h y ′ , z i : z ∈ aco { j ◦ N ( F ) : F ∈ Σ E } } = d sup s ( y ′ , aco { j ◦ N ( F ) : F ∈ Σ E } ) ≤ d max { s ( y ′ , conv { j ◦ N ( F ) : F ∈ Σ E } ) , s ( y ′ , conv {− j ◦ N ( F ) : F ∈ Σ E } ) } = d max { s ( y ′ , { j ◦ N ( F ) : F ∈ Σ E } ) , s ( y ′ , {− j ◦ N ( F ) : F ∈ Σ E } ) } ADON-NIKOD´YM REPRESENTATIONS FOR THE
BDS m -INTEGRAL 19 So, if { x ′ , . . . , x ′ n } ⊂ X ′ and a i ∈ R , i = 1 , . . . , n , then n X i =1 a i s ( x ′ i , M ( E )) = n X i =1 a i h e x ′ i , j ◦ M ( E ) i = (cid:28) n X i =1 a i e x ′ i , j ◦ M ( E ) (cid:29) ≤ d max (cid:26) s (cid:18) n X i =1 a i e x ′ i , { j ◦ N ( F ) : F ∈ Σ E } (cid:19) ,s (cid:18) n X i =1 a i e x ′ i , {− j ◦ N ( F ) : F ∈ Σ E } (cid:19)(cid:27) = d max (cid:26) sup F ∈ Σ E (cid:28) n X i =1 a i e x ′ i , j ◦ N ( F ) (cid:29) , sup F ∈ Σ E (cid:28) n X i =1 a i e x ′ i , − j ◦ N ( F ) (cid:29)(cid:27) = d max (cid:26) sup F ∈ Σ E (cid:20) n X i =1 a i h e x ′ i , j ◦ N ( F ) i (cid:21) , sup F ∈ Σ E (cid:20) − n X i =1 a i h e x ′ i , j ◦ N ( F ) i (cid:21)(cid:27) = d max (cid:26) sup F ∈ Σ E (cid:20) n X i =1 a i s ( x ′ i , N ( F )) (cid:21) , sup F ∈ Σ E (cid:20) − n X i =1 a i s ( x ′ i , N ( F )) (cid:21)(cid:27) = d max (cid:26) sup F ∈ Σ E (cid:20) n X i =1 a i s ( x ′ i , N ( F )) (cid:21) , − inf F ∈ Σ E (cid:20) n X i =1 a i s ( x ′ i , N ( F )) (cid:21)(cid:27) = d sup aco (cid:26) n X i =1 a i s ( x ′ i , N ( F )) : F ∈ Σ E (cid:27) Since we have also − n X i =1 a i s ( x ′ i , M ( E )) ≤ d sup aco (cid:26) n X i =1 a i s ( x ′ i , N ( F )) : F ∈ Σ E (cid:27) , we deduce that n X i =1 a i s ( x ′ i , M ( E )) ∈ d aco (cid:26) n X i =1 a i s ( x ′ i , N ( F )) : F ∈ Σ E (cid:27) . (5.6.jjj) ⇒ (5.6.j): By the assumption ± n X i =1 a i s ( x ′ i , M ( E )) ≤ d sup aco (cid:26) n X i =1 a i s ( x ′ i , N ( F )) : F ∈ Σ E (cid:27) ≤ d sup aco (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 a i s ( x ′ i , N ) (cid:12)(cid:12)(cid:12)(cid:12) ( F ) : F ∈ Σ E (cid:27) ≤ d (cid:12)(cid:12)(cid:12)(cid:12) n X i =1 a i s ( x ′ i , N ) (cid:12)(cid:12)(cid:12)(cid:12) ( E ) . Hence, for every set E ∈ Σ we have (cid:12)(cid:12)(cid:12)(cid:12) n X i =1 a i s ( x ′ i , M )( E ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ d (cid:12)(cid:12)(cid:12)(cid:12) n X i =1 a i s ( x ′ i , N ) (cid:12)(cid:12)(cid:12)(cid:12) ( E )and the standard calculation gives the required inequality for the variations. (5.6.j) ⇒ (5.6.jj): is obvious. (5.6.jj) ⇒ (RN j ): Assume that for each ε > ε/ > δ > P nm =1 a m x ′ m ∈ span B X ′ and each E ∈ Σ , we have (cid:12)(cid:12)(cid:12)(cid:12) n X m =1 a m s ( x ′ m , N ) (cid:12)(cid:12)(cid:12)(cid:12) ( E ) ≤ δ, ⇒ (cid:12)(cid:12)(cid:12)(cid:12) n X m =1 a m s ( x ′ m , M ) (cid:12)(cid:12)(cid:12)(cid:12) ( E ) ≤ ε/ . It follows that(20) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:28) n X m =1 a m e x ′ m , j ◦ N (cid:29)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( E ) ≤ δ ⇒ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:28) n X m =1 a m e x ′ m , j ◦ M (cid:29)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( E ) ≤ ε/ . Let E ∈ Σ and y ′ ∈ ℓ ′∞ ( B X ′ ) be such that |h y ′ , j ◦ N i| ( E ) ≤ δ/
2. Then, let E + y ′ ∪ E − y ′ = E be the Hahn decomposition of E with respect to h y ′ , j ◦ N i .Since { e x ′ : x ′ ∈ B X ′ } is norming for ℓ ∞ ( B X ′ ) there exists z ′ ∈ span { e x ′ : x ′ ∈ B X ′ } suchthat |h y ′ − z ′ , j ◦ N ( E + y ′ ) i| < δ/ |h y ′ − z ′ , j ◦ N ( E − y ′ ) i| < δ/ |h y ′ − z ′ , j ◦ M ( E + y ′ ) i| < δ/ |h y ′ − z ′ , j ◦ M ( E − y ′ ) i| < δ/ |h z ′ , j ◦ N i ( E ) | = |h z ′ , j ◦ N ( E + y ′ ) i + h z ′ , j ◦ N ( E − y ′ ) i| = [ h z ′ , j ◦ N ( E + y ′ ) i − h y ′ , j ◦ N ( E + y ′ ) i ]+ [ h z ′ , j ◦ N ( E − y ′ ) i − h y ′ , j ◦ N ( E − y ′ ) i ]+ [ h y ′ , j ◦ N ( E + y ′ ) i + h y ′ , j ◦ N ( E − y ′ ) i ] < δ. By the assumption |h z ′ , j ◦ M i| ( E ) ≤ ε/
2. Now we follow the reverse way: |h y ′ , j ◦ M i| ( E ) = h y ′ , j ◦ M ( E + y ′ ) i + h y ′ , j ◦ M ( E − y ′ ) i = [ h y ′ , j ◦ M ( E + y ′ ) i − h z ′ , j ◦ M ( E + y ′ ) i ]+ [ h y ′ , j ◦ M ( E − y ′ ) i − h z ′ , j ◦ M ( E − y ′ ) i ]+ [ h z ′ , j ◦ M ( E + y ′ ) i + h z ′ , j ◦ M ( E − y ′ ) i ] < δ + ε/ < ε. That means that j ◦ M is s - usac with respect to j ◦ N and so, in virtue of Theorem 5.1 j ◦ M has the derivative.The proof of the local versions is analogous to that given in Theorem 4.5 where the construction isgiven for the general case of an arbitrary multimeasure. (cid:3) Remark 5.7.
Theorems 5.1 and 5.6 allow us to observe that j ◦ M ≪ j ◦ N if and only if M is s - usac with respect to N . Analogously j ◦ M is usd by j ◦ N if and only if M is s - usd by N . Remark 5.8.
It turns out that the condition ( RN j ) implies the following one (that coincides withthe sub condition in case of vector measures): (sub): ∀ E ∈ Σ M ( E ) ⊂ d aco [ R ( N E ) ∪ R ( − N E )] . In fact, by Theorem 5.1 j ◦ M is subordinated to j ◦ N . Assume that d > E ∈ Σ , j ◦ M ( E ) ∈ d aco { j ◦ N ( F ) : F ∈ Σ E } ∈ cb ( ℓ ∞ ( B X ′ )) . This means that for every y ′ ∈ ℓ ′∞ ( B X ′ ), we have h y ′ , j ◦ M ( E ) i ≤ d s ( y ′ , aco { j ◦ N ( F ) : F ∈ Σ E } ) = d s ( y ′ , aco { j ◦ N ( F ) : F ∈ Σ E } ) ≤ d max { s ( y ′ , co { j ◦ N ( F ) : F ∈ Σ E } ) , s ( y ′ , co {− j ◦ N ( F ) : F ∈ Σ E } ) } = d max { s ( y ′ , { j ◦ N ( F ) : F ∈ Σ E } ) , s ( y ′ , {− j ◦ N ( F ) : F ∈ Σ E } ) } ADON-NIKOD´YM REPRESENTATIONS FOR THE
BDS m -INTEGRAL 21 In particular, if x ′ ∈ X ′ , then s ( x ′ , M ( E )) = h e x ′ , j ◦ M ( E ) i ≤ d s ( e x ′ , aco { j ◦ N ( F ) : F ∈ Σ E } )= d max { s ( e x ′ , { j ◦ N ( F ) : F ∈ Σ E } ) , s ( e x ′ , {− j ◦ N ( F ) : F ∈ Σ E } ) } = d max { s ( e x ′ , { j ◦ N ( F ) : F ∈ Σ E } ) , s {h e x ′ , − j ◦ N ( F ) i : F ∈ Σ E } ) } = d max { s ( x ′ , N ( F )) : F ∈ Σ E } ) , s {h x ′ , − N ( F ) i : F ∈ Σ E } ) } = d max { s ( x ′ , R ( N E ) ) , s ( x ′ , R ( − N E ) ) } ≤ d s ( x ′ , R ( N E ) ∪ R ( − N E ) ) ≤ d s ( x ′ , aco [ R ( N E ) ∪ R ( − N E )] ) = s ( x ′ , d aco [ R ( N E ) ∪ R ( − N E )] ) . In particular, if x ∈ M ( E ), then h x ′ , x i ≤ s ( x ′ , d aco [ R ( N E ) ∪ R ( − N E )] ). In virtue of the Hahn-Banach theorem x ∈ d aco [ R ( N E ) ∪ R ( − N E )]. Thus, M ( E ) ⊂ d aco [ R ( N E ) ∪ R ( − N E )] . The example given in Remark 4.6 proves that in case of arbitrary multimeasures the condition sub is sometimes essentially weaker than ( RN j ). The current example shows that also in caseof pointless multimeasures the sub does not guarantee the existence of the Radon-Nikod´ym de-rivative. It is enough to take a pointless N : Ω → cwk ( X ) and define M : Σ → cwk ( X ) by M ( E ) := aco [ R ( N E ) ∪ R ( − N E )].Next theorem provides a Radon-Nikod´ym representation without invoking to the R˚adstr¨om embed-ding. Theorem 5.9.
Let
M, N : Σ → cb ( X ) be two consistent d H -multimeasures. If M is s-usac (s-usdor s-uss) with respect to N , then there exists a non-negative, measurable, bounded, BDS m -integrablewith respect to N function θ : Ω → R such that M ( E ) = Z E θ dN, ∀ E ∈ Σ. Proof.
By Theorem 5.6 the s - usac ( s - usd , s - uss ) condition on M and N is equivalent to property( RN j ). Then, for every x ′ ∈ B X ′ , it is h e x ′ , j ◦ M ( E ) i = Z E θ d h e x ′ , j ◦ N i which is equivalent to s ( x ′ , M ( E )) = Z E θ ds ( x ′ , N ) . Then, by Proposition 2.8, θ is non-negative N -a.e.. Now, by (RN j ) and Theorem 5.2, we have that θ is BDS m -integrable with respect to N and M ( E ) = Z E θdN. (cid:3) Corollary 5.10.
Let
M, N : Σ → cb ( X ) satisfy the hypotheses of Theorem 5.9. If j ◦ M can berepresented as a BDS-integral with respect to j ◦ N with a non-negative bounded density θ , then also M can be represented as the BDS m -integral of θ with respect to N . Examples
Example 6.1.
Let µ be a finite measure on ( Ω, Σ ) and N ( E ) := [0 , µ ( E )] be an interval valuedmultimeasure (for results and applications of this kind of multimeasure see for example [10, 28] andthe references therein). We can observe that if f is a scalar, bounded measurable function, then f is BDS m -integrable with respect to N and M f ( E ) = (cid:20) − Z E f − dµ, Z E f + dµ (cid:21) . Example 6.2.
Let let N be as in Example 6.1 and let now M ( E ) := [0 , ν ( E )] where ν is a finitemeasure on ( Ω, Σ ), equivalent with µ . Let ν ( E ) = Z E θ dµ for all E ∈ Σ . Then there exists apartition Ω = S n Ω n such that 0 ≤ θ ≤ n µ -a.e. on Ω n .Moreover let ξ : R → R be defined by: ξ ( a ) = a if a >
0, otherwise ξ ( a ) = 0. So for every a ∈ R s ( a, N ( E )) = ξ ( a ) µ ( E ). If α, β ∈ R and x ′ = a, y ′ = b ∈ R then | αs ( a, N ) + βs ( b, N ) | ( E ) = | αξ ( a ) + βξ ( b ) | µ ( E ) and | αs ( a, M ) + βs ( b, M ) | ( E ) = | αξ ( a ) + βξ ( b ) | ν ( E ) . It follows that if E ∈ Σ Ω n , then | αs ( a, M ) + βs ( b, M ) | ( E ) = | αs ( a, N ) + βs ( b, N ) | ( E ) · ν ( E ) µ ( E ) ≤ n | αs ( a, N ) + βs ( b, N ) | ( E ) . Therefore the multimeasure M is locally usd with respect to the multimeasure N and can be repre-sented as a BDS m -integral with respect to N . One can easily check that M ( E ) = Z E θ dN . Example 6.3.
Assume that X is a Banach space and consider ([0 , , L , λ ) where λ is Lebesguemeasure and L is the family of all Lebesgue measurable subsets of [0 , f, g : [0 , → X betwo Pettis integrable functions and Γ( t ) := co { , f ( t ) } , ∆( t ) := co { , g ( t ) } be the multifunctionsdetermined by f and g respectively. They are ck ( X )-valued and Pettis integrable (see [6, Propositions2.3 and 2.5]). We observe that, for every x ′ ∈ X ′ , s ( x ′ , Γ( t )) = h x ′ , f i + ( t ) , s ( x ′ , ∆( t )) = h x ′ , g i + ( t ) . (21)If M, N : L → cwk ( X ) are the indefinite multivalued Pettis integrals of f, g , then ∀ x ′ ∈ X ′ , ∀ E ∈ L , s ( x ′ , M ( E )) = Z E h x ′ , f i + dλ, & s ( x ′ , N ( E )) = Z E h x ′ , g i + dλ. (22)Assume that there exists a measurable scalar function θ which is BDS m -integrable with respect to N and M ( E ) = Z E θ dN, ∀ E ∈ L . Since
M, N are positive, θ is non-negative. It is a consequence of Definition 2.3 that s ( x ′ , M ( E )) = Z E θ ds ( x ′ , N ) for all x ′ ∈ X ′ and E ∈ L . Due to (22) we have Z E h x ′ , f i + dλ = Z E θ ds ( x ′ , N ) = Z E θ h x ′ , g i + dλ for all x ′ ∈ X ′ and E ∈ L . It follows that for each x ′ ∈ X ′ we have h x ′ , f i = θ h x ′ , g i µ -a.e. Example 6.4.
Assume that X is a Banach space and µ is a non-trivial atomless finite measure on( Ω, Σ ). Let f , f : Ω → X be scalarly integrable functions and r , r : Ω → (0 , ∞ ) be µ -integrablefunctions. Following [24] or [6, Example 2.13], we define for i = 1 , Γ i : Ω → cb ( X )by the formulae Γ i ( ω ) = B ( f i ( ω ) , r i ( ω )), where B ( x, δ ) is the closed ball with its center in x and ofradius δ . Then s ( x ′ , Γ i ( ω )) = h x ′ , f i ( ω ) i + r i ( ω ) k x ′ k and so each Γ i is scalarly integrable. If f , f arePettis integrable, then each Γ i is Pettis integrable in cb ( X ). Moreover,( P ) Z E Γ i dµ = B (cid:18) ( P ) Z E f i dµ, Z E r i dµ (cid:19) i = 1 , . ADON-NIKOD´YM REPRESENTATIONS FOR THE
BDS m -INTEGRAL 23 Let M ( E ) := ( P ) Z E Γ dµ and N ( E ) := ( P ) Z E Γ dµ . One can easily check that M and N are d H -measures. Suppose that M ( E ) = Z E θ dN, E ∈ Σ , i.e. ∀ x ′ ∈ X ′ , ∀ E ∈ Σ, s ( x ′ , M ( E )) = Z E θ + ds ( x ′ , N ) + Z E θ − ds ( x ′ , − N ) . That yields the equality ∀ x ′ ∈ X ′ , ∀ E ∈ Σ, Z E h x ′ , f i dµ + k x ′ k Z E r dµ = Z E θ h x ′ , f i dµ + k x ′ k Z E r | θ | dµ . But the sets (cid:26)Z E f i dµ : E ∈ Σ (cid:27) , i = 1 , = x ′ ∈ X ′ vanishing on these sets. It follows that r = r | θ | µ -a.e. Appealing to [24, Lemma] we find that foreach x ′ ∈ X ′ one has h x ′ , f i = θ h x ′ , f i µ -a.e. (the exceptional set depends on x ′ ) i.e. f is scalarlyequivalent to θ · f .One can easily check that also the reverse implication holds true: if there exists a measurable function θ such that f is scalarly equivalent to θf and r = r | θ | µ -a.e., then M = Z θ dN .A similar calculation shows that j ◦ M can be represented as a BDS -integral with respect to j ◦ N ifand only if the above θ is non-negative.If we assume only that r and R are only positive and measurable, then we are in the local versionof the RN-theorem. Example 6.5.
If we assume in Example 6.4 that X = Z ′ and f , f are only Gelfand integrablethen Γ , Γ are weak ∗ multimeasures (see [24] for definitions). Applying now Remark 4.7, we see that ∀ z ∈ Z, ∀ E ∈ Σ, s ( z, M ( E )) = Z E θ + ds ( z, N ) + Z E θ − ds ( z, − N )if and only there is θ such that r = r · | θ | µ -a.e. and f is weak ∗ scalarly equivalent to θ · f . Acknowledgement
The first and the last authors are members of the working group Teoria dell’Approssimazione e Applicazioniof the Italian Mathematical Union (U.M.I.) and they were partially supported by: • Grant “Analisi reale, teoria della misura ed approssimazione per la ricostruzione di immagini”(2020) of GNAMPA – INDAM (Italy) • University of Perugia – Fondo Ricerca di Base 2019.
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Luisa Di Piazza : Department of Mathematics, University of Palermo, Via Archirafi 34, 90123 Palermo, (Italy). Email:[email protected], Orcid ID: 0000-0002-9283-5157
Kazimierz Musia l : Institut of Mathematics, Wroc law University, Pl. Grunwaldzki 2/4, 50-384 Wroc law, (Poland).Email: [email protected], Orcid ID: 0000-0002-6443-2043