BOUNDED SETS IN THE RANGE OF AN X

Vol. 23, No. 1 (2000) 21–30 S0161171200001708

© Hindawi Publishing Corp.

BOUNDED SETS IN THE RANGE OF AN X

-VALUED MEASURE WITH BOUNDED VARIATION

B. MARCHENA and C. PIÑEIRO (Received20 July 1998)

Abstract.LetXbe a Banach space andA⊂Xan absolutely convex, closed, and bounded set. We give some sufficient and necessary conditions in order thatAlies in the range of a measure valuedin the bidual spaceX^∗∗andhaving boundedvariation. Among other results, we prove thatX^∗ is a G. T.-space if andonly ifAlies inside the range of some X^∗∗-valuedmeasure with boundedvariation whenever XA is isomorphic to a Hilbert space.

Keywords and phrases. Banach spaces, Banach discs, vector measure, bounded variation.

2000 Mathematics Subject Classification. 46G10, 47B10.

1. Introduction. IfXis an infinite dimensional Banach space, it is well known that its unit ball cannot lie inside the range of anX^∗∗-valuedmeasure with boundedvari- ation. This paper is devoted to a study of bounded sets in a Banach space having that property. Since the convex andclosedhull of the range of a measure is the range of another one (see Diestel and Uhl [3]), we can reduce to consider bounded subsets which are closedandabsolutely convex. Such a set is calleda Banach disc. IfA⊂Xis a Banach disc, recall that∪^∞_i=1nAis a vector space andthat it can be endowedwith the norm

ρA(x)=inf

λ >0 :x∈λA

for allx∈ ∪^∞

i=1nA. (1.1)

It is easy to prove that(∪^∞_i=1nA,ρA)is a Banach space which will be denoted byXA

andthatAis the unit closedball of this space.jAwill denote the canonical map from XAintoX. SincejA(A)=A, it follows thatjAis a bounded linear map (see Junek [4]).

First of all, we obtain characterizations of linear operatorsT :X→Y that take the unit ball ofX into a subset ofY lying in the range of someY-valued(respectively, Y^∗∗-valued) measure with bounded variation. Concretely, we prove that operators belonging toᐃ◦Π^d₁(X,Y )take the unit ballBXinto a set lying in the range of some Y-valuedmeasure with boundedvariation. Nevertheless, we show that there exist operators satisfying that property but they do not belong to the classᐃ◦Π^d₁.

Next we give some sufficient and necessary conditions in order that a Banach disc AinXlies in the range of someX^∗∗-valuedmeasure with boundedvariation. Under certain conditions aboutAwe prove thatXAmust be isomorphic to a Hilbert space.

When XA is isomorphic to a Hilbert space we say that A is a Hilbert disc. Among other results, we show that the Banach spaces whose dual spaces are G. T.-spaces

are the only Banach spaces for which every Hilbert disc lies inside the range of some X^∗∗-valuedmeasure with boundedvariation.

Finally, we consider a class of subsets ofΠ1(X,Y )which we call(µ,c)-uniformly dominated. Such a set is a subsetMofΠ1(X,Y )with the property

T x ≤c

B_X∗

x,x^∗dµ(x^∗) for allx∈X,T∈M, (1.2)

where c is a positive constant and µ is a probability measure on (BX^∗,-weak).

Questions regarding the finer structure of these sets have found interest since Grothendieck-Pietsch’s discovery of their famous Domination theorem. We prove that Mis uniformly dominatedif andonly if there exists someX^∗-valuedmeasuremwith bounded variation such that

T^∗(BY^∗)⊂rg(m) for allT∈M. (1.3)

We use the classical notation in Banach spaces theory. We consider all Banach spaces over real numbers. IfXis a Banach space,X^∗ will denote its dual space and BX its closedunit ball. For a subsetKofX, aco(K)will be the absolutely convex andclosed hull ofK.

We consider only countably additive measures defined on σ-algebras. If Σ is a σ-algebra of subsets of a setΩ,Xis a Banach space andm:Σ→Xis such a measure, we denote by|m|the total variation ofm. If|m|is finite, we say thatmhas bounded variation. The range ofmis denoted by rg(m), that is, rg(m)= {m(A):A∈Σ}.

2. Banach operators taking the unit ball inside the range of some vector measure.

Following Piñeiro [6], we denote by ᏾_bv(X,Y )the vector space of linear operators taking compact subsets ofXinto sets that lie inside the range of aY-valuedmeasure with boundedvariation. The author provedthat an operatorT belongs to᏾bv if and only if the adjoint mapT^∗:Y^∗→X^∗is 1-summing. Next we prove that, in fact, these operators take the unit ball ofXinside the range of someY^∗∗-valuedmeasure with bounded variation.

Theorem2.1. LetXandY be Banach spaces. Suppose thatT:X→Y is a bounded operator.T maps the unit ballBXin a set lying in the range of aY^∗∗-valued measure with bounded variation if and only if the adjoint mapT^∗is1-summing.

Proof. (⇒)LetT:X→Y be an operator such thatT (BX)lies inside the range of aY^∗∗-valuedmeasure with boundedvariation. Obviously, the mapiY◦T belongs to Rbv(X,Y^∗∗)andby Piñeiro [6] the map(iY◦T )^∗is 1-summing. ThenT^∗:Y^∗→X^∗is 1-summing too.

(⇐)LetT :X→Y be an operator such that its adjoint is 1-summing. In light of Grothendieck-Pietsch’s Domination theorem (see Diestel et al. [2]) there is a regular Borel probability measureµon (BY^∗∗, weak^∗) such thatT^∗:Y^∗→X^∗factors through a subspaceHofL¹(µ)in the way

23 Y^{∗ T∗}

//

C

B B B B B B !!

B B

H,

D

== |

| |

| |

| |

|

whereC andD are bounded linear maps and ˜C:Y^∗→L¹(µ)is integral. By duality we have

X ^D

∗0

//

//

L^∞(µ),

φ

OO

C˜^∗

:: v

v v v v v v

v v

whereD₀^∗ denotes the restriction map ofD^∗ toX and φ is the quotient map. ˜C^∗ is integral andω^∗-ωcontinuous. So, if we define aY^∗∗-valuedmeasure bym(E)= C˜^∗(xE) for all Borel setE in (BY^∗∗, weak^∗), mis a countably additive Y^∗∗-valued measure having bounded variation (see Diestel-Uhl [3]). By Piñeiro [6], there is another Y^∗∗-valuedmeasure ˜msuch that ˜C^∗(BL^∞(µ))=rg(m)˜ and|m| ≤˜ 2|m|. Finally, we have

T (BX)=C^∗◦D₀^∗(BX)⊂ D₀^∗ C^∗(BH^∗)

= D₀^∗ C^∗

φ(BL^∞(µ))

⊂ D^∗₀ C^∗

φ(BL^∞(µ))

= D₀^∗ C˜^∗(BL^∞(µ))= D^∗₀ C˜^∗(BL^∞(µ))= D₀^∗ rg(m).˜

(2.3)

Now, we give a characterization of operatorsT:X→Y taking the unit ball ofXinside the range of someY-valuedmeasure with boundedvariation.

Theorem2.2. LetXandY be Banach spaces andT:X→Yan operator.T (BX)lies inside the range of someY-valued measure with bounded variation if and only ifT^∗ factors through a subspaceHof anL¹(µ)-space

Y^{∗ T∗}

//

A

J J J J J %%

J J

J J

H⊂L¹(µ),

B

99 t

t t t t t t

t t

whereµ is a positive and finite measure,A:Y^∗→L¹(µ)is integral and-weak-weak continuous, andB:H→X^∗is a bounded operator.

Proof. We only prove the “if part” because the “only if part” is similar to the above theorem. LetT:X→Y be an operator such thatT (BX)⊂rg(m),m:Σ→Y is a vector measure with bounded variation. We denote byµthe variation measure ofm.

The integration operatorI:L^∞(µ)→Y defined byI(f )=

f dmfor allf∈L^∞(µ)is 1-summing and, therefore, integral (see Diestel-Uhl [3]). SoI^∗:Y^∗→L^∞(µ)^∗is integral, but

I^∗(y^∗)=d(y^∗◦m)

dµ ∈L¹(µ) for ally^∗∈Y^∗. (2.5)

Then the rank ofI^∗is containedinL¹(µ)andthe map A:y^∗∈Y^∗ →d(y^∗◦m)

dµ ∈L¹(µ) (2.6)

is-weak-weak continuous andintegral. Finally, let

H= {Ay^∗:y^∗∈Y^∗}^L1(µ), (2.7) anddefineB:H→X^∗byB(Ay^∗)=T^∗y^∗. SinceT (BX)⊂rg(m), it is easy to prove thatBis well definedandcontinuous.

Remark2.3. (a) In [5], it is showedthat there exist operators taking the unit ball in the range of someY^∗∗-valuedmeasure with boundedvariation, but they do not take it in the range of someY-valuedmeasure of boundedvariation.

(b) Suppose thatT1:X→Y is an operator whose adjoint mapT₁^∗is 1-summing and T2:Y →Z is a weakly compact operator. Then, Theorem 2.2 tells us that the map T2◦T1takes the unit ball ofXinto a subset ofZlying in the range of someZ-valued measure with bounded variation. Now, we prove that the converse is not true, i.e., there exist operators taking the unit ball inside the range of some vector measure with bounded variation but they do not belong to the classᐃ◦᏾bv (hereᐃdenotes the operator ideal of all weakly compact operators). To see this, we need the following result.

Lemma2.1. Let T = T2◦T1 where T1∈ ᏾_bv(X,Y ) and T2∈ ᐃ(Y ,Z). Then the sequence(T xn)lies inside a countable sum of segments whenever(xn)is a bounded sequence inX.

Recall that a countable sum of segments in a Banach spaceXis a set of the form ∞

n=1

[−ωn,ωn]=



 ∞ n=1

αnωn:(αn)∈B*∞



, (2.8)

where

ωnis an absolutely convergent series inX. Such a set is obviously the range of a vector measure with boundedvariation. Piñeiro [7] provedthat a sequence(xn) lies inside a countable sum of segments if and only if the operator

(αn)∈*1 → ^∞

n=1

αnxn∈X (2.9)

is nuclear.

Proof. Given a bounded sequence(xn)inX, we define the operator S:(αn)∈*1 →

∞ n=1

αnT xn∈Z, (2.10)

whereS is the composition of T0:(αn)∈*1→ΣαnT1xn ∈NY and T2:Y →Z. It follows from Theorem 2.1 that the sequence(T1xn)lies inside the range of some Y^∗∗-valuedmeasure with boundedvariation. By [7, Lem 2]T0 is integral. Now by Grothendieck’s theorem (see Diestel and Uhl [3]) the compositionT2◦T0is nuclear.

25 Now we give the counterexample. Anantharaman andDiestel [1] showedthat the unit ball of*2is the range of ac0-valuedmeasure of boundedvariation. Nevertheless, the canonical basis(en)does not lie in a countable sum of segments inc0because the identity operator from*1intoc0is not nuclear.

3. Banach discs lying in the range of some X^∗∗-valued measure. Piñeiro and Rodriguez-Piazza [8] proved that only finite dimensional Banach spacesXhave the property that every compact subset is containedin the range of someX-valuedmea- sure with bounded variation. Then a natural question arises: given an infinite dimen- sional Banach spaceX, which bounded subsets ofXhave the following property (P)?

(P)“Every compact subset ofAis contained in the range of anX-valued measure with bounded variation”

We have obtainedthe following results:

Theorem3.1. LetXbe a Banach space. IfA⊂Xis a Banach disc with property (P), thenAis contained in the range of someX^∗∗-valued measure with bounded variation.

Proof. IfAis a Banach disc having property (P), it is obvious that the operator jA :XA→X maps every compactK⊂A(compact inXA) into a set lying inside the range of aX-valuedmeasure with boundedvariation. Again the theorem of Piñeiro [6]

tells us that (jA)^∗ :X^∗ →(XA)^∗ is 1-summing. Theorem 2.1 concludes the proof.

Remark. By Theorem 2.1, a setA⊂X lies inside the range of someX^∗∗-valued measure with boundedvariation if andonly if (jA)^∗ :X^∗ →(XA)^∗ is 1-summing.

According to Grothendieck-Pietsch’s theorem, there exist a regular Borel positive mea- sureµon (BX^∗∗,-weak) such that

(jA)^∗(x^∗)≤

B_X∗∗

x^∗,x^∗∗dµ for allx^∗∈X^∗. (3.1) On the other hand, we have

(jA)^∗(x^∗)=sup

a∈A

a,(jA)^∗(x^∗)=sup

a∈A

a,x^∗. (3.2)

So, (3.1) can be written in the form supa∈A|a,x^∗| ≤

B_X∗∗|x^∗,x^∗∗|dµ for allx^∗∈X^∗. (3.3) Note that (3.3) implies that the operator

x^∗∈X^∗ →

a,x^∗

a∈A∈*^∞(A) (3.4)

is integral. So we have obtainedthe following result.

Theorem3.2. LetXbe a Banach space andA⊂X a Banach disc. The following statements are equivalent:

(i) The adjoint operator ofjA:XA→Xis 1-summing.

(ii) The operator(ψa)∈*¹(A)→Σa∈Aψaa∈Xis integral.

(iii) Alies inside the range of someX^∗∗-valued measure with bounded variation.

As usual “◦” denotes the polar in the dualityX,X^∗, “•” the polar in the duality X^∗,X^∗∗andX_A^∗◦the vector spaceX_/p^∗−1

A◦{0}endowed with the norm ˜PA^◦(x^∗+P_A⁻¹◦{0})

=PA^◦(x^∗), for allx^∗∈X^∗ (see Junek [4]). Recall thatPA^◦(x^∗)=supx∈A|x,x^∗|for allx^∗∈X^∗. In general,X_A^∗◦is not complete.

Theorem3.3. LetXbe a Banach space andA⊂Xa Banach disc such thatX_A^∗◦ is a Banach space. IfAis contained in the range of someX^∗∗-valued measure with bounded variation, then the Banach spaceXAis isomorphic to a Hilbert space.

Proof. We first prove that(X^∗∗)A^◦•is isomorphic to a Hilbert space. As we have mentionedearlier, there is a regular Borel positive measureµon (BX^∗∗, weak^∗) satis- fying (3.1). We may define onX^∗a scalar product(·/·)by letting

x^∗/y^∗

=

B_X∗∗x^∗,x^∗∗y^∗,y^∗∗dµ for allx^∗,y^∗∈X^∗. (3.5) We denote byp(·)the associate seminorm defined by

p(x^∗)=

B_X∗∗

x^∗,x^∗∗2dµ _1/2

for allx^∗∈X^∗. (3.6) By Hölder’s inequality we have

B_X∗∗

x^∗,x^∗∗dµ≤µ(BX^∗∗)^1/2p(x^∗) for allx^∗∈X^∗. (3.7)

Therefore, (3.3) and(3.7) yields supa∈A

a,x^∗≤µ(BX^∗∗)^1/2p(x^∗)≤µ(BX^∗∗) x^∗ for allx^∗∈X^∗. (3.8) Now we consider the vector spaceX_/p^∗−1{0}endowed with the scalar product

x^∗+p⁻¹{0}

y^∗+p⁻¹{0}

= x^∗/y^∗

for allx^∗,y^∗∈X^∗. (3.9) We can define a linear map J : X^∗_/p−1{0} →X_A^∗◦, J(x^∗+p⁻¹{0})= x^∗+p⁻¹_A◦{0} for allx^∗∈X^∗. From (3.8) it follows thatJ is well definedandcontinuous. Obviously, it is a surjection. ThenX_A^∗◦ is isomorphic to a quotient of the prehilbertian space X_/p^∗−1{0}. This implies that(XA^∗◦)^∗ is isomorphic to a subspace of the Hilbert space (X_/p^∗−1{0})^∗. As(X_A^∗◦)^∗ and X_A^∗∗◦• are isometric (see Junek [4]), we have provedthat X_A^∗∗◦• is isomorphic to a Hilbert space. Finally, we show that XA is isomorphic to a subspace ofX_A^∗∗◦•. By the open map theorem it suffices to prove thatXA, endowed with the restriction ofpA^◦•toXA, is complete because we have the relation

pA^◦•(x)≤pA(x) for allx∈XA. (3.10) To see this, let(xn)be a Cauchy sequence inX_A^∗∗◦•for whichxn∈XAfor alln∈N. In particular,(xn)is bounded forpA^◦•. Then there is a constantc >0 such thatxn∈cA^◦•

for alln∈N. Clearly,(xn)converges inX_A^∗∗◦• to a limitx^∗∗∈cA^◦•, andtherefore, x^∗∗=limxninX^∗∗too. SinceXis closedinX^∗∗it follows thatx^∗∗∈X. This shows thatx^∗∗∈cA^◦•. AsA^◦•=A, the theorem is established.

27 Remark3.4. (a) It is known that the convergence for the norm · does not im- ply the convergence forpA. We are going to show an easy example. LetX=L¹[0,1]

andA=aco{ᐄ[k−1/n,k/n]: 1≤k≤n,n∈N}. Obviously,Ais containedin the range of someX-valuedmeasure of boundedvariation, thenjA∈᏾bv(XA,X). By Piñeiro’s theo- rem [6],jAmaps every null sequence fromXAinto a sequence lying in a countable sum of segments. But again Piñeiro [6] provedthat the null sequence(ᐄ[k−1/n,k/n)])n∈N,1≤k≤n

is not containedin any countable sum of segments. Then this sequence does not con- verge inXA.

We do not know whether every Banach disc A ⊂X lying in the range of some X^∗∗-valuedmeasure of boundedvariation has property (P).

(b) IfAis the unit ball of a closedsubspaceY ofX, thenXAis isometric toY and X_A^∗◦ is isometric to the quotient spaceX^∗/Y^⊥. SoX_A^∗◦ is a Banach space. In this case, we have the following complete result.

Theorem3.5. LetXbe a Banach space andYa closed subspace ofX.BYhas prop- erty (P) if and only ifBY lies inside the range of some X^∗∗-valued measure having bounded variation. If this is the case,Y is isomorphic to a Hilbert space.

(c) Unfortunately, Theorem 3.2 is not true whenX_A^∗◦ is not complete. For example, consider the setA=BL^∞[0,1] in L¹[0,1]. A is a Banach disc inL¹[0,1]that lies in the range of someL¹[0,1]-valuedmeasure of boundedvariation since the identity operator fromL^∞[0,1]intoL¹[0,1]is Pietsch integral (see Diestel andUhl [3]). ButXA

is isometric toL^∞[0,1], so it cannot be isomorphic to a Hilbert space.

From now on in this section we suppose that the space under consideration belongs to a particular class. To start, we consider the Hilbert case.

Theorem3.6. LetX be a Hilbert space andA⊂Xa Banach disc such thatXA is isomorphic to a Hilbert space. Then the setAhas property (P) if and only if

i∈I

supa∈A

ei

a

²<+∞ (3.11)

for some orthonormal basis(ei)i∈IinX.

Proof. LetA⊂Xbe a Banach disc for whichXAis isomorphic to a Hilbert spaceH.

IfJ:H→XAis an isomorphism and(ei)i∈Ian orthonormal basis inX, we have

i∈I

supa∈A

ei

a

²=

i∈I

(jA)^∗ei 2≤ (J^∗)^{−1 2}

i∈I

J^∗◦(jA)^∗ei 2, (3.12)

on the other hand,

i∈I

J^∗◦(jA)^∗ei 2≤ J^{∗ 2}

i∈I

(jA)^∗ei 2= J^{∗ 2}

i∈I

supa∈A

ei

a

². (3.13) This proves that the map (jA◦J)^∗:X→H is Hilbert-Schmidt if and only if (3.11) holds. AsJ is an isomorphism, it follows that(jA)^∗is 1-summing only in this case.

Corollary3.7. LetX be a Hilbert space andA= {Σαnxn:(αn)∈B*2}, where (xn) is a sequence belonging to *_ω²(X). The set A has property (P) if and only if Σ xn 2< +∞.

Proof. In fact, if(ei)i∈I is an orthonormal basis inX, by Parseval’s identity we have

i∈I

supa∈A

ei

a

²=

i∈I α∈Bsup_*2

_∞ ei n=1αnxn

²=

i∈I α∈Bsup_*2

^∞

n=1

αn

ei

xn

²

=

i∈I

∞ n=1

ei

xn

²= ∞ n=1

i∈I

ei

xn

²= ∞ n=1

xn 2.

(3.14)

Now we are going to obtain an interesting result in caseX^∗ is a G. T.-space. Recall that a Banach space is calleda G. T.-space ifΠ1(X,H)=ᏸ(X,H)for all Hilbert space H(see Pisier [9]).

Theorem3.8. LetXbe a Banach space. The following statements are equivalent:

(i) X^∗is a G. T.-space.

(ii) ABanach discA⊂Xis contained in the range of someX^∗∗-valued measure with bounded variation wheneverXAis isomorphic to a Hilbert space.

Proof. The implication (i)⇒(ii) follows directly from Theorem 2.1 and the defini- tion of G. T.-spaces. So, letXbe a Banach space satisfying (ii). By Piñeiro [7], we only needto prove thatΠ1(*1,X)=Ᏽ(*1,X). To this end, let(xn)be a sequence inXsuch that the operatorT :(αn)∈*1→Σαnxn ∈X is 1-summing. By [7], there exists a sequence(yn)∈*²_ω(X)such that

xn∈

αnyn:(αn)∈B*2

. (3.15)

PutA= {Σαnyn:(αn)∈B*2}. AsXAis isomorphic to a Hilbert space, by hypothesis Ais containedin the range of someX^∗∗-valuedmeasure with boundedvariation. In particular, so is(xn). Again by Piñeiro [7],T is integral.

Next theorem proves that inᏸ1-spaces a Hilbert disc has the property (P) if and only if it is containedin some countable sum of segments inX.

Theorem3.9. LetXbe anᏸ1space andA⊂Xa Hilbert disc. The following state- ments are equivalent:

(i) Ahas property (P).

(ii) jA:XA→Xis nuclear.

(iii) Ais contained in some countable sum of segments inX.

Proof. (i)⇒(ii) IfA⊂Xhas property (P), Theorem 3.1 tells us thatAlies inside the range of someX^∗∗-valuedmeasure with boundedvariation. By Theorem 2.1,(jA)^∗: X^∗→(XA)^∗ is 1-summing. SinceX^∗ is anᏸ∞-space it follows that(jA)^∗ is integral.

Finally,(jA)^∗ is nuclear because(XA)^∗ is a dual space with Radon-Nikodym prop- erty (XAis isomorphic to a Hilbert space). As Hilbert spaces have the approximation property,jAitself is nuclear (see Diestel andUhl [3]).

The other implications are obvious.

29 4. (µ,c)-uniformly dominated sets. We finish our work by obtaining characteriza- tion, in terms of ranges of vector measures, of(µ,c)-uniformly dominated subsets of Π1(X,Y ).

Theorem4.1. LetX andY be Banach spaces, andM a subset of Π1(X,Y ). M is uniformly dominated if and only if there exists someX^∗-valued measuremof bounded variation such that

T^∗(BY^∗)⊂rg(m) for allT∈M. (4.1) Proof. (⇒) LetM⊂Π1(X,Y )be a(µ,c)-uniformly dominated set, i.e., we have

T x ≤c

B_X∗

x,x^∗dµ(x^∗) for allx∈X, T∈M, (4.2)

whereµis a regular Borel probability measure onBX^∗. This yields supa∈A|x,a| ≤c

B_X∗|x,x^∗|dµ(x^∗) for allx∈X, (4.3) whereA= ∪T∈MT^∗(BY^∗). By (4.3), the operator

U:x∈X→ x,a

a∈A∈*∞(A) (4.4)

is 1-summing. So isU^∗∗:X^∗∗→*∞(A), in particular,U^∗∗is integral. Now it suffices to notice thatU^∗∗(x^∗∗)=(a,x^∗∗)a∈A. Since there is a projection fromX^∗∗∗ in X^∗, it follows from Theorem 3.2 thatAis containedin the range of someX^∗-valued measure of bounded variation.

Conversely, suppose M ⊂ Π1(X,Y ) is a set satisfying T^∗(BY^∗) ⊂ rg(m) for all T ∈ M, m being some X^∗-valuedmeasure with boundedvariation. Put A = aco(∪T∈M T^∗(BY^∗)). A is a Banach disc in X^∗ that lies inside the range of some X^∗-valuedmeasure with boundedvariation. According to Theorem 3.2, the operator x^∗∗∈X^∗∗→(a,x^∗∗)a∈A∈*∞(A)is integral. So isx∈X→(x,a)a∈A∈*∞(A).

By Grothendieck-Pietsch’s theorem there exists a regular Borel positive measureµon BX^∗such that

supa∈A

x,x^∗≤

B_X∗|x,x^∗|dµ(x^∗) for allx∈X. (4.5) This yields

T x ≤

B_X∗|x,x^∗|dµ(x^∗) for allx∈XandallT∈M. (4.6) References

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Marchena: Departamento de Matemáticas, Escuela Politécnica Superior, Universi- dad de Huelva,21810La Rábida, Huelva, Spain

E-mail address:[email protected]

Piñeiro: Departamento de Matemáticas, Escuela Politécnica Superior, Universidad de Huelva,21810La Rábida, Huelva, Spain

E-mail address:[email protected]