Some properties and applications of equicompact sets of operators

STUDIA MATHEMATICA 181 (2) (2007) Some p ope ies and applica ions o equicompac se s o ope a o s by E. Se ano,C. Pi˜ nei o and J. M. Delgado (Huel a) Abs ac . Le Xand Ybe Banach spaces. A subse M o K(X, Y ) ( he ec o space o all compac ope a o s om Xin o Yendowed wi h he ope a o no m) is said o be equicompac i e e y bounded sequence (xn) in Xhas a subsequence (xk(n))nsuch ha (T xk(n))nis uni o mly con e gen o T∈M. We s udy he ela ionship be ween his con- cep and he no ion o uni o mly comple ely con inuous se and gi e some applica ions. Among o he esul s, we ob ain a gene aliza ion o he classical Ascoli heo em and a com- pac ness c i e ion in Mc(F, X), he Banach space o all ( ini ely addi i e) ec o measu es (wi h compac ange) om a ield Fo se s in o Xendowed wi h he semi a ia ion no m. 1. In oduc ion. Th oughou his pape Xand Ywill be Banach spaces. As usual, we will deno e by K(X, Y ) he Banach space o all com- pac ope a o s om Xin o Yendowed wi h he ope a o no m. In [9] he au ho s in oduced he no ion o an equicompac se o ope a o s. A se M⊂K(X, Y ) is said o be equicompac i e e y bounded sequence (xn) in Xhas a subsequence (xk(n))nsuch ha (Txk(n))nis uni o mly con e gen o T∈M. They p o ed ha he no ions o equicompac se and collec i ely compac se a e dual in he ollowing sense: M ⊂K(X, Y ) is equicompac ( espec i ely, collec i ely compac ) i M∗={T∗:T∈M}is collec i ely compac ( espec i ely, equicompac ). We ecall ha M is called collec i ely compac i he se ST∈MT(BX) is ela i ely compac . Thus, he well known Palme heo em [7] akes he ollowing new o m: Theo em A. I Mis a subse o K(X, Y ), hen he ollowing s a emen s a e equi alen : (i) M is ela i ely compac . (ii) M is equicompac and Mx={Tx :T∈M}is ela i ely compac o e e y x∈X. 2000 Ma hema ics Subjec Classi ica ion: 47B07, 46G10. Key wo ds and ph ases: compac ope a o s, equicompac se s o ope a o s, collec i ely compac se , ec o measu es, Ascoli’s heo em. [171] c Ins y u Ma ema yczny PAN, 2007 172 E. Se ano e al. (iii) M is collec i ely compac and M∗y∗={T∗y∗:T∈M}is ela i ely compac o e e y y∗∈Y∗. In pa icula , he au ho s o [9] ha e ob ained he ollowing cha ac e i- za ion o compac ness in a dual Banach space ha we will use h oughou his pape . Co olla y B. Le Xbe a Banach space and A⊂X∗a bounded se . Then Ais ela i ely compac i e e y bounded sequence (xn)in Xhas a subsequence (xk(n))nso ha (hxk(n), ai)nis uni o mly con e gen o a∈A. In [9], he au ho s p o ed ha a se M ⊂K(X, Y ) is equicompac i he e exis s a null sequence (x∗ n) in X∗such ha kTxk ≤ supn|hx, x∗ ni| o all x∈Xand T∈M. They also p o ed ha equicompac se s a e uni o mly comple ely con inuous, ha is, kTxnk → 0 uni o mly o T∈M whene e (xn) is a weakly null sequence in X. Ac ually, i he Banach space Xdoes no con ain a copy o ℓ1, equicompac se s and uni o mly comple ely con inuous se s a e he same (see P oposi ion 2.2 below). In his pape we deepen he s udy o he ela ionships be ween equicom- pac and uni o mly comple ely con inuous se s. Mo eo e , we ob ain a gen- e aliza ion o he classical Ascoli heo em and a cha ac e iza ion o compac - ness in Mc(F, X), he Banach space o all ( ini ely addi i e) ec o measu es om Fin o Xwi h compac ange, Fbeing a ield o subse s o a se Ω. We use he classical no a ion in Banach space heo y. I Xis a Banach space, X∗deno es i s dual space, BXi s closed uni ball and SXi s uni sphe e. Fo a subse Ao X,co(A) is he closed con ex hull o A. As usual, ℓ1(I, X) ( espec i ely ℓ∞(I, X)) s ands o he Banach space o all unc ions bx:I→Xsa is ying Pi∈Ikbx(i)k<∞( espec i ely sup{kbx(i)k:i∈I}<∞) endowed wi h i s na u al no m. We will use he ollowing e sion o he well known Vala compac ness c i e ion in ℓc ∞(I, X) ( he Banach space o all unc ions bx:I→Xwi h ela i ely compac ange endowed wi h he sup emum no m). Theo em 1.1 (K. Vala [10, Theo em 1]).Le M⊂ℓc ∞(I, X)be bounded. The ollowing s a emen s a e equi alen : (i) M is ela i ely compac . (ii) M has he ollowing p ope ies: (a) Fo e e y ε > 0 he e exis s a ini e pa i ion {D1,...,Dp}o I such ha 1≤k≤p, i, j ∈Dk⇒ kbx(i)−bx(j)k< ε o all bx∈M. (b) M(i) = {bx(i) : bx∈M}is ela i ely compac o all i∈I. Ou no a ion om ec o measu e heo y ollows [3]. We only conside ec o measu es de ined on ields o se s. I Fis a ield o subse s o a se Ω, Equicompac se s o ope a o s 173 Xis a Banach space and m:F→Xis such a measu e, we deno e by kmk(A) he semi a ia ion o A∈F: kmk(A) = sup{|x∗◦m|(A) : x∗∈BX∗}. The ange o mis deno ed by g(m), ha is, g(m) = {m(A) : A∈F}. Finally, we deno e by B(F) he Banach space o all scala - alued unc ions on Ω ha a e uni o m limi s o simple unc ions modeled on F. 2. Rela ionships be ween equicompac and uni o mly comple- ely con inuous se s Theo em 2.1.Le Mbe a bounded subse o K(X, Y ). The ollowing s a emen s a e equi alen : (i) M is equicompac . (ii) M has he ollowing p ope ies: (a) M is uni o mly comple ely con inuous. (b) Fo e e y semino malized sequence (xn)in Xequi alen o he ℓ1uni ec o basis and e e y ε > 0, he e exis s a ini e pa i ion {D1,...,Dp}o Nsuch ha kTxn−Txmk< ε o all T∈M whene e m, n ∈Diand i= 1,...,p. P oo . (i)⇒(ii). We only ha e o p o e (b). Le φ:ℓ1→span{xn:n∈N} be an isomo phism wi h φ(en) = xn o all n∈N. Ob iously, he se M◦φ= {T◦φ:T∈M}is equicompac and, he e o e, φ∗◦M∗is collec i ely compac . Tha is, he se [ T∈M φ∗(T∗(BY∗) = {(hxn, T∗y∗i) : y∗∈BY∗, T ∈M} is ela i ely compac in ℓ∞. Acco ding o Theo em 1.1, gi en ε > 0, he e exis s a ini e pa i ion {D1,...,Dp}o Nso ha n, m ∈Di⇒ |hxn−xm, T∗y∗i| < ε o all y∗∈BY∗and T∈M, o i= 1,...,p. This yields kTxn−T xmk< ε o all T∈M and i= 1,...,p. (ii)⇒(i). Le (xn) be a bounded sequence in X. By Rosen hal’s ℓ1- heo em, (xn) has a subsequence which is ei he weakly Cauchy o equi alen o he uni basis o ℓ1( o simplici y, we will go on deno ing i by (xn)). In he i s case, (Txn) is uni o mly con e gen o T∈M because M is uni o mly comple ely con inuous. In he second case, by hypo hesis, he e exis s a pa i ion {D1,...,Dp}o Nso ha n, m ∈Di⇒ kTxn−Txmk<1 o all T∈M and i= 1,...,p. Some o he Di’s mus be in ini e, so we can choose i≤p such ha Diis in ini e. I k1:N→Diis an inc easing bijec ion, hen (xk1(n)) is a sequence equi alen o he uni basis o ℓ1; so epea ing his p ocess 174 E. Se ano e al. induc i ely, we can de e mine a sequence o subsequences (kp(n))nsuch ha (kp+1(n))nis a subsequence o (kp(n))nand kTxkp(n)−Txkp(m)k<1/p o all n, m ∈Nand T∈M o all p∈N. Now i is easy o deduce ha (Txkp(p))pis uni o mly con e gen o T∈M. The nex p oposi ion p o es ha all uni o mly comple ely con inuous se s a e equicompac i Xdoes no con ain a copy o ℓ1. We deno e by V(X, Y ) he ec o space o all comple ely con inuous ope a o s om X in o Yendowed wi h he ope a o no m. P oposi ion 2.2.Le Xbe a Banach space. The ollowing s a emen s a e equi alen : (i) Fo e e y Banach space Yand e e y M⊂V(X, Y ), M is equicom- pac whene e Mis uni o mly comple ely con inuous. (ii) The e exis s a Banach space Ysuch ha e e y uni o mly comple ely con inuous se M⊂K(X, Y )is equicompac . (iii) Xdoes no con ain copy o ℓ1. P oo . I X6←֓ ℓ1, hen V(X, Y ) = K(X, Y ) o all Banach spaces Y and (iii)⇒(i) can be deduced using Theo em 2.1; so we only ha e o p o e (ii)⇒(iii). Assuming (ii), o p o e ha Xdoes no con ain a copy o ℓ1, we show ha e e y uni o mly comple ely con inuous subse Ao X∗is ela i ely compac [5, Th. 2]. Take y0∈SYand pu M = A⊗y0. I is ob ious ha M is a uni o mly comple ely con inuous subse o K(X, Y ). So, by hypo hesis, M is equicompac , which yields he equicompac ness o Aas a subse o K(X, R). Finally, a call o Co olla y B ells us ha Ais ela i ely compac . Recall ha an ope a o T:X→Yis said o be condi ionally weakly compac i e e y bounded sequence (xn) in Xadmi s a subsequence (xk(n))n so ha (Txk(n))nis weakly Cauchy. We deno e by CW(X, Y ) he ec o space o all condi ionally weakly compac ope a o s om Xin o Y. P oposi ion 2.3.Fo an ope a o Q∈L(X, Z), he ollowing s a e- men s a e equi alen : (i) Q∈CW(X, Z). (ii) I A⊂Z∗is uni o mly comple ely con inuous, hen Q∗(A)is ela- i ely compac . (iii) Fo e e y Banach space Yand e e y N⊂V(Z, Y ), N◦Qis equicom- pac whene e Nis uni o mly comple ely con inuous. P oo . (i)⇒(ii). Le A⊂Z∗be uni o mly comple ely con inuous and Q∈CW(X, Z). Gi en a bounded sequence (xn) in X, he e exis s a subse- quence (xk(n))nsuch ha (Qxk(n))nis weakly Cauchy. Then (hQxk(n), ai)n= Equicompac se s o ope a o s 175 (hxk(n), Q∗ai)nis uni o mly con e gen o a∈A. Now, Co olla y B con- cludes he p oo . (ii)⇒(iii). Le Ybe a Banach space and N ⊂V(Z, Y ) uni o mly com- ple ely con inuous. We p o e ha Q∗◦N∗is collec i ely compac . Fo his, ake a sequence ((Q∗◦S∗ n)y∗ n)nin SS∈NQ∗◦S∗(BY∗) and pu A={S∗ ny∗ n: n∈N}. The se Ais uni o mly comple ely con inuous. In ac , i (zn) is a weakly null sequence in Z, we ha e |hzn, S∗ my∗ mi| =|hSmzn, y∗ mi| ≤ kSmznk. Then, by hypo hesis, he se Q∗(A) is ela i ely compac and, he e o e, ((Q∗◦S∗ n)y∗ n)nhas a con e gen subsequence. (iii)⇒(i). By hypo hesis, S(Q(BX)) is ela i ely compac o all Banach space Yand all S∈V(Z, Y ). Acco ding o [8, p. 377], he se Q(BX) is condi ionally weakly compac . The nex heo em shows ha e e y equicompac se M admi s a ep e- sen a ion o he o m M = N◦Q, whe e N is uni o mly comple ely con inuous and Qis condi ionally weakly compac . Theo em 2.4.Le Mbe a subse o L(X, Y ). The ollowing s a emen s a e equi alen : (i) M is equicompac . (ii) The e exis a closed subspace Zo c0,Q∈K(X, Z)and N⊂ K(Z, Y )such ha Nis equicompac and M = N ◦Q. (iii) The e exis a Banach space Z,Q∈CW(X, Z)and N⊂V(Z, Y ) such ha Nis uni o mly comple ely con inuous and M = N ◦Q. P oo . Only (i)⇒(ii) needs o be p o ed. Acco ding o [9, P op. 2.2], he equicompac ness o M implies ha he e exis s a null sequence (x∗ n) in X∗ so ha kTxk ≤ sup n |hx, x∗ ni| o all x∈Xand T∈M. Fo each n∈N, we de ine λn=pkx∗ nkand b∗ n=λ−1 nx∗ n(we can assume ha λn6= 0 o all n∈N). Ob iously, λn→0 and kb∗ nk → 0. Now, in a simila way o he p oo o [6, Th. 17.1.4], we ind a closed subspace Z o c0and ope a o s Q∈K(X, Z) and ST∈K(Z, Y ) sa is ying T=ST◦Q, o all T∈M (Q:x∈X7→ (hx, b∗ ni)∈c0,Z={Qx :x∈X}and ST(hx, b∗ ni) = Tx). Pu N = {ST:T∈M}. Since Z ֒→c0, we ha e Z∗≈ℓ1/Z⊥. I (en) deno es he uni ec o basis o ℓ1, i is clea ha h(hx, b∗ ni), λm[em]i=hx, λmb∗ mi=hx, x∗ miand kλn[en]k → 0. 176 E. Se ano e al. Then, o all T∈M, we ha e kST(hx, b∗ ni)k=kTxk ≤ sup m |hx, x∗ mi| = sup m |h(hx, b∗ ni), λm[em]i|, ha is, N is equicompac . Finally, no ice ha M = N ◦Q. 3. A gene aliza ion o he classical Ascoli heo em. In his sec ion we gene alize he no ion o equicompac se o a wide class o unc ions. Le Jbe an a bi a y se and Za comple e me ic space. I M is a se o unc ions om Jin o Zwi h ela i ely compac ange, we say ha M is equicompac i e e y sequence (jn) in Jhas a subsequence (jk(n))nsuch ha ( (jk(n)))nis uni o mly con e gen o ∈M. I M⊂K(X, Y ), whe e Xand Ya e Banach spaces, hen Mis equicom- pac (in he o iginal sense) i M = {T|BX:T∈M}is equicompac . Th oughou his sec ion Xwill be a Banach space and Ian in ini e in- dex se . The mapping ψ:bx∈ℓc ∞(I, X)7→ ψ(bx) = Tbx∈K(ℓ1(I), X) de- ined by Tbx(ξi)i∈I=Pi∈Iξibx(i) is an isome ic isomo phism. Using a sim- ila a gumen o he p oo o Theo em 2.1, i is easy o p o e he nex lemma: Lemma 3.1.Le Mbe a bounded subse o K(ℓ1(I), X). Then Mis equicompac i o e e y ε > 0 he e exis s a ini e pa i ion {D1,...,Dp} o Isuch ha 1≤k≤p, i, j ∈Dk⇒ kTei−Tejk< ε o all T∈M. Rema k 3.2.I ψ(M) ⊂K(ℓ1(I), X) is equicompac , hen M is equicom- pac and bounded, bu , in gene al, an equicompac se M in ℓc ∞(I, X) is no necessa ily bounded. To see his, ake an equicompac and bounded se- quence (bxk) in ℓc ∞(I, X) and choose x0∈SX. Now, o each k∈N, deno e by bzk∈ℓc ∞(I, X) he unc ion de ined by bzk(i) = bxk(i) + kx0 o all i∈I. I is easy o p o e ha (bzk) is an equicompac sequence bu , ne e heless, i is no bounded. P oposi ion 3.3.Le Mbe a bounded subse o ℓc ∞(I, X). The ollow- ing s a emen s a e equi alen : (i) M is equicompac . (ii) ψ(M) is equicompac . (iii) The e exis s a null sequence (b βn)in ℓ∞(I)so ha kbx(i)−bx(j)k ≤ sup n |b βn(i)−b βn(j)| o all i, j ∈Iand bx∈M. P oo . (i)⇒(ii). Conside he ope a o U:ℓ1(I)→ℓ∞(M, X) de ined by U(ei) = (bx(i))bx∈M o all i∈I((ei)i∈Iis he canonical basis o ℓ1(I)). By (i), Uis compac and, he e o e, he se ψ(M) is equicompac [9, P op. 2.2]. Equicompac se s o ope a o s 177 (ii)⇒(iii). Acco ding o [9, P op. 2.2], he e exis s a null sequence (b βn) in ℓ∞(I) sa is ying kTbx(ξ)k ≤ supn|hξ, b βni| o all ξ∈ℓ1(I) and bx∈M. In pa icula , o i, j ∈Iwe ha e kbx(i)−bx(j)k ≤ supn|b βn(i)−b βn(j)| o all bx∈M. (iii)⇒(i). Gi en a sequence (in) in I, he e exis s a subsequence (ik(n))n such ha (heik(n),b βmi)nis uni o mly con e gen o m∈Nbecause o he compac ness o {b βm:m∈N}(Co olla y B). Now, om he uni o m con- e gence o (b βm(ik(n)))n o m∈Nand (iii), i is easy o ob ain (i). Rema k 3.4.Acco ding o Lemma 3.1 and P oposi ion 3.3, a bounded subse M o ℓc ∞(I, X) is equicompac i i sa is ies condi ion (ii)(a) in Vala’s heo em (Th. 1.1). As usual, i Ωis a compac opological space, C(Ω, X) is he Banach space o all con inuous unc ions φ:Ω→Xendowed wi h he sup emum no m. Ob iously, C(Ω, X) is a subspace o ℓc ∞(Ω, X). Now we a e eady o show he main esul o his sec ion: a gene aliza ion o he classical Ascoli heo em [4, Th. 7.5.7]. Theo em 3.5.Le Mbe a subse o C(Ω, X), Ωbeing an a bi a y compac opological space. The ollowing s a emen s a e equi alen : (i) M is ela i ely compac . (ii) M has he ollowing p ope ies: (a) M is equicompac . (b) M(ω) = {φ(ω) : φ∈M}is ela i ely compac o all ω∈Ω. P oo . (i)⇒(ii). ollows di ec ly om Theo em 1.1, P oposi ion 3.3 and Lemma 3.1. Acco ding o Rema k 3.4, o p o e (ii)⇒(i) we only need o show ha M is bounded. To see his, conside he unc ion F:ω∈Ω7→ F(ω) = (φ(ω))φ∈M∈ℓ∞(M, X). Then Fis well de ined and has compac ange since M is equicompac . The nex p oposi ion lis s some elemen a y p ope ies o equicompac se s o unc ions ha allow us o conside he abo e heo em as a gene al- iza ion o he classical Ascoli–A zel`a heo em. P oposi ion 3.6.Le Ωbe an a bi a y compac opological space and Ma bounded subse o C(Ω, X). (1) I Mis equicompac , hen i is sequen ially equicon inuous. (2) I ,in addi ion,Ωis me izable, hen Mis equicompac i i is equicon inuous. P oo . (1) Suppose (ωn) is a sequence in Ωwi h limi ω0∈Ω. By con- inui y, o each φ∈M, we ha e φ(ωn)→φ(ω0). As M is equicompac , by con adic ion, i is easy o p o e ha φ(ωn)→φ(ω0) uni o mly in φ∈M. 178 E. Se ano e al. (2) In case Ωis me izable, equicon inui y and sequen ial equicon inui y a e he same. So, we only ha e o p o e he su iciency. Assume M is equicon- inuous. Gi en a sequence (ωn) in Ω, as Ωis me izable and compac , he e is a con e gen subsequence (ωk(n))n. Suppose ha ωk(n)→ω0∈Ω. Since M is equicon inuous i ollows ha φ(ωk(n))→φ(ω0) as n→ ∞ uni o mly in φ∈M. 4. Compac ness in Mc(F, X).As in Sec ion 3, we say ha a se M ⊂ Mc(F, X) is equicompac i e e y sequence (An) in Fhas a subsequence (Ak(n))nsuch ha (m(Ak(n)))nis uni o mly con e gen o m∈M. By [3, I.5.3], all ec o measu es in Mc(F, X) a e s ongly addi i e. To s a , we p o e ha equicompac se s o ec o measu es a e uni o mly s ongly addi i e. We ecall ha a se M o s ongly addi i e ec o measu es is called uni o mly s ongly addi i e i , o e e y sequence (An) o pai wise disjoin membe s o F, limn→∞ kP∞ k=nm(Ak)k= 0 uni o mly in m∈M. P oposi ion 4.1.I Mis an equicompac se o ec o measu es, hen i is uni o mly s ongly addi i e. P oo . Acco ding o [3, P oposi ion I.1.17], we ha e o p o e ha limn→∞ km(An)k= 0 uni o mly in m∈M whene e (An) is a sequence o pai wise disjoin membe s o F. A guing by con adic ion, suppose he e exis ε > 0, a sequence (mn) in M and a subsequence (Ak(n))nso ha (1) kmn(Ak(n))k> ε o all n∈N. By hypo hesis, (Ak(n))nhas a subsequence (Ah(n))nsuch ha limnm(Ah(n)) = 0 uni o mly in m∈M, which con adic s (1). I m:F→Xis a ini e addi i e measu e wi h compac ange, hen he in eg a ion map Im: ∈B(F)7→ T Ω dm ∈Xis compac . In ac , in [3, p. 263] i is p o ed ha he sums o he o m Pn i=1 αim(Ai), 0 ≤α1≤ · · · ≤ αn≤1, Ai∩Aj=∅ o i6=j, belong o co( g(m)). This yields he inclusion n Ω dm : ∈BB(F)o⊂co( g(m)) −co( g(m)). Then he ope a o Imis compac . Now we a e eady o s a e ou main esul . Theo em 4.2.Le Mbe a subse o Mc(F, X). The ollowing s a emen s a e equi alen : (i) M is ela i ely compac . (ii) M is equicompac and M(A)is ela i ely compac o all A∈F. P oo . (i)⇒(ii). Pu b M = {Im:m∈M}. As he e is an isome y be- ween Mc(F, X) and K(B(F), X) de ined by m↔Im,b M is a ela i ely Equicompac se s o ope a o s 179 compac subse o K(B(F), X). By Theo em A, b M is equicompac and b M( ) = { T Ω dm :m∈M}is ela i ely compac , o all ∈B(F). So, in pa icula , (ii) holds. (ii)⇒(i). We conside he ec o measu e G:A∈F7→ (m(A))m∈M∈ℓ∞(M, X), which has compac ange since M is equicompac . Thus he in eg a ion map IG:B(F)→ℓ∞(M, X) is compac and de ined by IG( ) = (Im( ))m∈M o ∈B(F). F om he compac ness o IGi ollows ha b M = {Im:m∈M} is equicompac . To p o e ha b M is ela i ely compac in K(B(F), X), we only ha e o show ha b M( ) = { T Ω dm :m∈M}is ela i ely compac o all ∈B(F). Gi en ∈B(F), choose a sequence (φn)no simple unc ions so ha = limn→∞ φnin B(F). Fix ε > 0, and ake n∈Nsuch ha k −φnk< ε/s, whe e s= sup{kmk(Ω) : m∈M}. Fo all m∈M, we ha e Ω dm = Ω ( −φn)dm + Ω φndm ∈M(φn) + εBX, since k T Ω( −φn)dmk ≤ ε o all m∈M. I is ob ious ha M(φn) is ela i ely compac , so we ha e p o ed ha b M( ) is ela i ely compac o all ∈B(F). Co olla y 4.3.Le (mn)be an equicompac sequence in Mc(F, X). I limn→∞ mn(A)exis s o all A∈F, hen (mn)is con e gen . Rema k 4.4.A uni o mly s ongly addi i e se is no necessa ily equi- compac . Fo an example, ake a noncompac and weakly compac subse Wo L1(µ), µbeing Lebesgue measu e on [0,1]. Deno e by M(W) he se o inde ini e in eg als λ = T (·) dµ wi h unning o e W. By [1, Th. VII.13], M(W) is uni o mly coun ably addi i e, ne e heless, i is no equicompac in iew o Theo em 4.2. Examples o equicompac se s can be ob ained in he ollowing way: ake a uni o mly comple ely con inuous se N ⊂V(Z, X), Zbeing an a bi a y Banach space, and a ec o measu e m∈Mc(F, X). I is easy o p o e ei he di ec ly o using P oposi ion 2.4 ha he se M = N ◦mis equicompac . Theo em 4.5.Le Mbe a bounded subse o Mc(F, X). Then Mis equicompac i he e exis a Banach space Z,a ec o measu e m∈Mc(F, X) and a uni o mly comple ely con inuous se N⊂V(Z, X)so ha M = N◦m. P oo . We only ha e o p o e he necessi y. So, le M ⊂Mc(F, X) be bounded and equicompac . As in he p oo o he abo e heo em, we can conside he ec o measu e G:A∈F7→ (m(A))m∈M∈ℓ∞(M, X).