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).