PORTUGALIAE MATHEMATICA
Vol. 55 Fasc. 1 – 1998
A NEW EXTENSION OF KOMLÓS’ THEOREM
IN INFINITE DIMENSIONS.
APPLICATION: WEAK COMPACTNESS IN L1X
M. Saadoune *
Abstract: A new version of Komlós’ theorem in infinite dimensions is presented.
It gives a different approach to well-known theorems (C. Castaing [11], A. Ülger [30] and
J. Diestel–W.M. Ruess–W. Schachermayer [17]).
1 – Introduction
The purpose of this paper is to prove Komlós’ theorem for bounded sequences
in L1X (X being a separable Banach space) that satisfy a tightness condition
formulated in the manner of A. Ülger [30]. We obtain our result (Theorems
1 and 2) by combining truncation arguments and the diagonal process based
on successively applying the scalar version of Komlós’ theorem, via the biting
lemma. Corollaries 3–6 are special cases of Theorems 1–2 and include results of
E.J. Balder [5].
As applications, we recover directly from Corollary 3, by using the Lebesgue–
Vitali theorem, a relative weak compactness criterion of C. Castaing [[11], Th. 4.1,
p. 2.14]: (Theorem 7). More generally, we obtain, from Theorem 1, a general
weak compactness criterion (Theorem 8): the relative weak compactness in L 1X
is characterized by the uniform integrability and the tightness condition used in
Theorem 1. This result involves recent criterions obtained by A. Ülger [30] and
J. Diestel–W.M. Ruess–W. Schachermayer [17].
Received : September 9, 1996; Revised : June 11, 1997.
Keywords: Bounded sequences in L1X , Biting lemma, Uniform integrability, Tightness,
Komlós’ theorem, Weak compactness.
* This paper was partially written while the author was visiting the University of Montpellier
II. He gratefully acknowledges the support of (Action Intégrée No 95/0849).
114
M. SAADOUNE
2 – Notations, definitions and background
Throughout this paper the triple (Ω, F, µ) is a finite measure space. Without
loss of generality we assume µ(Ω) = 1. (X, | · |) is a separable Banach space. The
norm topology on X will be referred to by the symbol s. The weak topology
σ(X, X ∗ ) on X will be indicated similarly by the symbol w. Here X ∗ stands for
the topological dual of X, the duality between X and X ∗ is denoted by h , i. The
(prequotient) space of all X-valued Bochner-integrable functions will be denoted
by L1X .
Recall (see [23]) that the dual of L1X is the (quotient) space L∞
X ∗ [X] of scalarly
∗
measurable bounded functions from Ω into X . We denote by L0X ∗ [X] the
(prequotient) space of scalarly measurable functions from Ω into X ∗ .
Also recall that a subset H of L1X is uniformly integrable (briefly UI) if
lim sup
t→∞ u∈H
Z
|u| dµ = 0 .
[|u|>t]
It is well known that H is UI if it is bounded (i.e. supu∈H
lim sup
µ(A)→0 u∈H
Z
R
Ω |u| dµ
< +∞) and
|u| dµ = 0 .
A
Definition. A set H of measurable functions is w-tight (resp. s-tight) if ∀² > 0,
there exists a measurable multifunction (for measurability of multifunctions see
[13]) with w-compact (resp. s-compact) values K² such that
∀ u ∈ H, µ
³n
ω ∈ Ω : u(ω) ∈
/ K² (ω)
o´
≤².
The following equivalent formulation of tightness is given in [4].
Definition. A set H of measurable functions is w-tight (resp. s-tight) if there
exists an F ⊗B-measurable integrand h : Ω×X → [0, +∞] which is w-inf-compact
(resp. s-inf-compact) and such that
(‡)
sup
u∈H
Z
h(ω, u(ω)) dµ < +∞ .
Ω
This notion of tightness goes back to 1979 (see C. Castaing [10] and K.T.
Andrews [2]). For more on tightness see [1], [7], [24], [27]. Note that every
bounded subset of L1X is w-tight whenever X is reflexive.
A NEW EXTENSION OF KOMLÓS’ THEOREM
115
Remark. In the first formulation of tightness the multifunctions K² can
be assumed to be convex valued but in the second formulation its not always
possible to replace h by a convex inf-compact function: the following is due to
M. Valadier. Let dω be the Lebesgue measure on [0, 1].
Proposition. Let X be a Banach space containing a bounded sequence
(en )n∈N∗ which does not admit any w-convergent subsequence. Then there exists
a bounded set H in L1X ([0, 1], dω) which is s-tight and such that there exists no
measurable integrand h : [0, 1] × X → [0, +∞] which is convex and w-inf-compact
in x and which satisfies (‡).
Proof: We may suppose without loss of generality that m 6= n ⇒ mem 6=
nen 6= 0. Let us consider H : ={nen 1A : n ∈ N∗ , µ(A) = 1/n} ∪ {0}. Then H
is bounded in L1X . Let h0 be defined by h0 (0) = 0, h0 (nen ) = n for n ≥ 1 and
h0 (x) = +∞ elsewhere. Thus h0 is s-inf-compact and satisfies
sup
u∈H
Z
[0,1]
(h0 ◦ u)(ω) dω ≤ 1 ,
hence H is s-tight.
Now suppose that h : [0, 1] × X → [0, +∞] is any convex weakly inf-compact
integrand satisfying (‡). Then h(ω, en ) → +∞ because otherwise there would
exist M < +∞ and a subsequence (enk )k such that supk h(ω, enk ) ≤ M and
the sequence (enk )k would admit a w-convergent further subsequence. If un =
nen 1An ∈ H,
Z
Z
[0,1]
h(ω, un (ω)) dω ≥
An
h(ω, nen ) dω .
R
It is possible to choose An such that [0,1] h(ω, un (ω)) dω → +∞. Indeed, since
0 ∈ H, one has h(ω, 0) < +∞ a.e. By convexity h(ω, nen ) ≥ n[h(ω, en ) −
(1 − n1 ) h(ω, 0)] ≥ n[h(ω, en ) − h(ω, 0)]. Then by Egorov’s theorem, there exists
a Borel subset B of [0, 1] such that µ(B) ≥ 1/2 and such that the sequence
(h(., en ) − h(., 0))n converges uniformly to +∞ on B. It remains to choose, for
n ≥ 2, An contained in B.
Let us introduce the following notion of convergence [7].
Definition. Let (un )n∈N∗ ∪{∞} be a sequence of measurable functions from
Ω into X and Y be a subset of X ∗ .
– (un )n is said to K-converge (resp. σ(X, Y )-K-converge) almost everywhere
on Ω to u∞ , if for every subsequence (uni )i of (un )n there exists a null set
116
M. SAADOUNE
N ∈ F such that for every ω ∈ Ω\N ,
n
1X
un (ω) → u∞ (ω)
n i=1 i
³
∗
resp. ∀ x ∈ Y,
D
n
E
´
1X
uni (ω) → hx∗ , u∞ (ω)i ;
x ,
n i=1
∗
– (un )n is said to K-converge in measure to u∞ , if for every subsequence (uni )i
of (un )n
n
1X
un → u∞ in measure .
n i=1 i
Remark.
Let (an )n∈N∗ be a sequence in R and a ∈ R. Then, (an )n
K-converges to a (consider each an as a constant function from Ω into R in
the definition above) is equivalent to an → a.
A celebrated discovery of Komlós [25] is as follows:
Theorem. Let (un )n be a bounded sequence of integrable functions from Ω
into R. Then (un )n has a subsequence which K-converges a.e. to an integrable
function.
This result has received a new proof by S.D. Chatterji [14], (see also [29] for
a short proof).
3 – Generalized Komlós’ theorem
The main result of this paper is the following theorem under a w-tightness
assumption.
Theorem 1. Let (un )n be a bounded sequence in L1X such that the following
tightness condition holds
T1 : Given any subsequence (uki )i of (un )n , there exists a w-tight sequence
(vn )n with vn ∈ co{uki : i ≥ n}.
Let D be a countable σ(X ∗ , X)-dense subset of the unit ball B ∗ of X ∗ . Then
there exist a function u∞ in L1X and a subsequence (u0n )n of (un )n such that
(a) (u0n )n σ(X, D)-K-converges a.e. to u∞ ;
117
A NEW EXTENSION OF KOMLÓS’ THEOREM
(b1 ) every w-tight sequence (vn )n , with vn ∈ co{u0i : i ≥ n} has a subsequence
(vn0 )n such that
(vn0 )n
∀h ∈
σ(X, D)-K-converges a.e. to u∞ ,
L0X ∗ [X] ,
(hh, vn0 i)n
K-converges in measure to hh, u∞ i .
Proof: The proof will be given in three steps.
Step 1 – Existence of (u0n )n : by the biting lemma (see [18], a proof of the biting
lemma is reproduced in [26]), there exist an increasing sequence of measurable
sets Ap with limp→∞ µ(Ap ) = 1 and a subsequence (u0n )n of (un )n such that
∀ p, the sequence (1Ap u0n )n is UI .
(1)
On the other hand, using Komlós’ theorem and the diagonal sequence process,
we obtain the existence of integrable functions αx∗ , (x∗ ∈ D), from Ω into R,
and of a subsequence of (u0n )n , which we may still denote by (u0n )n , such that for
every further subsequence (u0ni )i of (u0n )n
(2)
∀ x∗ ∈ D ,
n
1X
hx∗ , u0ni (ω)i → αx∗ (ω)
n i=1
a.e.
Let (vn )n be an arbitrary fixed and w-tight sequence, with vn ∈ co{u0i : i ≥ n}.
Then, for each q ∈ N∗ , there exists a measurable multifunction with w-compact
values Kq such that
(3)
∀n,
where
µ(Ω\Bn,q ) ≤
n
1
,
q
o
Bn,q : = ω ∈ Ω : vn (ω) ∈ Kq (ω) .
The same diagonal process used above gives for the sequences (hx∗ , vn i)n and
(hx∗ , 1Bn,q vn i)n , the existence of integrable functions βx∗ and βx∗ ,q (x∗ ∈ D,
q ∈ N∗ ), and of a subsequence (vn0 )n of (vn )n such that for every further subsequence (vn0 i )i of (vn0 )n
n
1X
hx∗ , vn0 i (ω)i → βx∗ (ω)
n i=1
(4)
∀ x∗ ∈ D ,
(5)
∀ x∗ ∈ D, ∀ q ∈ N∗ ,
a.e. ,
n
1X
hx∗ , 1Bn0 ,q vn0 i (ω)i → βx∗ ,q (ω)
i
n i=1
a.e. ,
0 ) is the subsequence of (B
where for each q ∈ N∗ , (Bn,q
n
n,q )n corresponding to
0
(vn )n .
118
M. SAADOUNE
By (1), (2), (4) and Vitali’s theorem we have, for every x∗ ∈ D, every p, every
A ∈ Ap ∩ F, every subsequence (u0ni )i of (u0n )n and every subsequence (vn0 i )i of
(vn0 )n
Z
n Z
1X
lim
hx∗ , u0ni i dµ =
αx∗ dµ
n→∞ n
A
i=1 A
and
n
1X
lim
n→∞ n
i=1
Z
hx
A
∗
, vn0 i i dµ
=
Z
A
βx∗ dµ .
Hence, by using the remark given in section 2, we get, for every x∗ ∈ D, every p
and every A ∈ Ap ∩ F,
lim
(6)
Z
hx∗ , u0n i dµ =
Z
Z
hx∗ , vn0 i dµ =
Z
n→∞ A
lim
(7)
n→∞ A
A
A
αx∗ dµ ,
βx∗ dµ .
Each vn0 is of the form
vn0 =
X
λi u0i+n ,
with λi ≥ 0 and
X
λi = 1 .
i≤Jn
i≤Jn
Using (6), (7) and this expression of vn0 , one can easily see that for every x∗ ∈ D,
every p and every A ∈ Ap ∩ F,
Z
A
αx∗ dµ =
Z
A
βx∗ dµ .
Therefore, since D is countable and limp→∞ µ(Ap ) = 1, we deduce that for every
x∗ ∈ D
(8)
αx∗ (ω) = βx∗ (ω)
a.e.
Step 2 – Existence of u∞ : we define for all n ∈ N∗ and q ∈ N∗ the following
partial sums
n
n
1X
1X
0
:=
vi0 and Sn,q
1 0 v0 .
Sn0 : =
n i=1
n i=1 Bi,q i
1) Let q ∈ N∗ . We may assume that Kq (ω) is convex and contains 0 for each
ω ∈ Ω by considering the corresponding multifunction co(Kq (.) ∪ {0}), thanks
to Krein and Eberlein–Šmulian’s theorem [[22], Th. 19 E]. The measurability
of this new map follows from [[13], Th. III.40] and [[21], Remark (1), p. 163].
A NEW EXTENSION OF KOMLÓS’ THEOREM
119
0 (ω) belongs to the w-compact set K (ω) for all n ∈ N∗ and all ω ∈ Ω,
Since Sn,q
q
0 (ω)) (possibly depending upon
there exists a subsequence (Sn0 i ,q (ω))i of (Sn,q
n
ω), which w-converges to some limit point xq,ω . But, because of (5) the entire
0 (ω)) w-converges to the same limit point whenever ω is outside
sequence (Sn,q
n
some negligible set N . Thus, we obtain
(9)
∀ x∗ ∈ D, ∀ ω ∈ Ω\N ,
βx∗ ,q (ω) = hx∗ , xq,ω i a.e.
∞
Define u∞
q (ω) : = xq,ω for ω ∈ Ω\N and uq (ω) : = 0 for ω ∈ N . Then it is easily
seen by the Pettis measurability theorem [[16], Th. 2, II] that u∞
q is measurable.
0
2) Let h ∈ L∞
X ∗ [X]. Since the sequence (hh, 1Ap Sn,q i)n is UI and converges
∗
a.e. to hh, 1Ap u∞
q i for all p and all q ∈ N , the Lebesgue–Vitali theorem gives
Z
(10)
Ap
¯
¯
¯
¯
0
i − hh, u∞
¯hh, Sn,q
q i¯ dµ → 0 ,
hence
∀ p, ∀ q ∈ N∗ ,
0
→ 1 Ap u ∞
1Ap Sn,q
q
weakly in L1X .
Furthermore, ∀ p, ∀ q ∈ N∗ ,
sup
n
Z
Ap
|Sn0
−
0
Sn,q
| dµ
n
1X
≤ sup
n n
i=1
≤ sup
n
Z
0 )
Ap ∩(Ω\Bi,q
0 )
Ap ∩(Ω\Bn,q
≤ sup sup
n
Z
m
Z
|vi0 | dµ
|vn0 | dµ
0 )
Ap ∩(Ω\Bn,q
|u0m | dµ .
As (1) and (3) ensure
∀p,
lim sup sup
q→∞ n
m
Z
0 )
Ap ∩(Ω\Bn,q
|u0m | dµ = 0 ,
we obtain
(11)
∀p,
lim sup
q→∞ n
Z
Ap
0
| dµ = 0 .
|Sn0 − Sn,q
Consequently, by Grothendieck’s weak relative compactness lemma [[20], ch. 5,
4, n◦ 1], for each p, the sequence (1Ap Sn0 )n is relatively weakly compact in L1X .
120
M. SAADOUNE
3) For each p choose a weak limit point u∞,p of the sequence (1Ap Sn0 )n . Then,
by (7), we obtain ∀ x∗ ∈ D, ∀ A ∈ Ap ∩ F,
Z
A
βx∗ dµ = lim
Z
hx∗ , vn0 i dµ
= lim
Z
hx∗ , Sn0 i dµ
n→∞ A
=
n→∞ A
Z
A
hx∗ , u∞,p i dµ ,
hence
∀ x∗ ∈ D ,
βx∗ (ω) = hx∗ , u∞,p (ω)i
a.e. in Ap .
Since limp→∞ µ(Ap ) = 1, there exists a measurable function u∞ defined on all Ω
such that
∀ x∗ ∈ D ,
(12)
βx∗ (ω) = hx∗ , u∞ (ω)i
a.e.
Moreover, by (4) and (12), we have
∀ x∗ ∈ D ,
hx∗ , Sn0 (ω)i → hx∗ , u∞ (ω)i
a.e.
which implies
|u∞ (ω)| ≤ lim inf |Sn0 (ω)|
a.e.
n→∞
and hence
Z
Ω
|u∞ | dµ ≤
Z
lim inf |Sn0 | dµ .
Ω n→∞
By Fatou’s lemma we get
Z
Ω
|u∞ | dµ ≤ sup
n
Z
Ω
|Sn0 | dµ
≤ sup
n
Z
Ω
|un | dµ < +∞
and hence u∞ ∈ L1X .
Step 3 – (u0n )n and u∞ have the required properties.
1) By combining (2), (4), (8) and (12), we obtain
(13)
(u0n )n and (vn0 )n
σ(X, D)-K-converge a.e. to u∞ .
Furthermore, from (5) and (9) we deduce that
(14)
∀ q ∈ N∗ ,
0
∞
∗
0 v )n σ(X, X )-K-converges a.e. to u
(1Bn,q
n
q .
121
A NEW EXTENSION OF KOMLÓS’ THEOREM
Fix h in L0X ∗ [X]. We shall show that (hh, vn0 i)n K-converges in measure to hh, u∞ i.
By considering the function h/(1+|h|), we may suppose that |h|∞ < 1. Let (vn0 i )i
be a subsequence of (vn0 )n . By (13) and (14), we have
∀ x∗ ∈ D ,
hx∗ , Sn00 (ω)i → hx∗ , u∞ (ω)i
a.e.
and
∀ x∗ ∈ D, ∀ q ∈ N∗ ,
where
Sn00 : =
00
hx∗ , Sn,q
(ω)i → hx∗ , u∞
q (ω)i
n
n
1X
1X
00
:=
vn0 i and Sn,q
1B 0 v 0
n i=1
n i=1 ni ,q ni
a.e.
(n ∈ N∗ ) .
Then
∀ x∗ ∈ D, ∀ q ∈ N∗ ,
¯D
E¯
¯
¯ ∗ 00
00
(ω) ¯ →
¯ x , Sn (ω) − Sn,q
Hence for all q ∈ N∗ and for a.e. ω we have
¯D
E¯
¯
¯ ∗
¯ x , u∞ (ω) − u∞
q (ω) ¯
a.e.
¯
¯
¯D
E¯
¯
¯
¯ ∗
¯
(ω)
=
sup
¯u∞ (ω) − u∞
¯
¯ x , u∞ (ω) − u∞
q
q (ω) ¯
∗
x ∈D
¯
¯
¯
¯
00
≤ lim inf ¯Sn00 (ω) − Sn,q
(ω)¯ .
n
This inequality implies that ∀ n ∈ N∗ , ∀ p and ∀ q ∈ N∗
Z
Ap
Z
¯
¯
¯
¯
00
¯hh, Sn i−hh, u∞ i¯ dµ ≤
Z ¯
¯
¯
¯
¯
¯
¯
00
00 ¯
00
i−hh, u∞
i
¯hh, Sn i−hh, Sn,q i¯ dµ +
¯hh, Sn,q
q ¯ dµ
Ap
Ap
Z ¯
¯
¯
¯
+
¯hh, u∞
q i − hh, u∞ i¯ dµ
Ap
≤
Z
Ap
+
Z
00
|Sn00 − Sn,q
| dµ +
lim inf
n
Ap
|Sn00
−
Z
Ap
¯
¯
¯
¯
00
i − hh, u∞
i
¯hh, Sn,q
q ¯ dµ
00
Sn,q
| dµ
.
Then by Fatou’s lemma, ∀ n ∈ N∗ , ∀ p and ∀ q ∈ N∗ ,
Z
Ap
Z
¯
¯
¯
¯
00
hh,
S
i−hh,
u
i
dµ
≤
2
sup
¯
∞ ¯
n
n
Ap
00
|Sn00 − Sn,q
| dµ +
Z
Ap
¯
¯
¯
¯
00
i−hh, u∞
¯hh, Sn,q
q i¯ dµ .
Using similar arguments to those used to prove (10) and (11), we obtain
∗
∀ p, ∀ q ∈ N ,
lim
Z
n→∞ Ap
¯
¯
¯
¯
00
i − hh, u∞
¯hh, Sn,q
q i¯ dµ = 0
122
M. SAADOUNE
and
∀p,
lim sup
q→∞ n
hence
∀p,
lim
Z
n→∞ Ap
Z
Ap
00
| dµ = 0
|Sn00 − Sn,q
¯
¯
¯
¯
¯hh, Sn00 i − hh, u∞ i¯ dµ = 0 .
Let ² > 0. For p and n sufficiently large, we have
µ(Ω\Ap ) <
²
2
Z
and
Ap
¯
¯
²2
¯
¯
.
¯hh, Sn00 i − hh, u∞ i¯ dµ ≤
2
Therefore, by Markov’s inequality, we get
µ
then
µ
µ½
µ½
¾¶
¯
¯
²
¯
¯
00
≤
ω ∈ Ap : ¯hh(ω), Sn (ω)i − hh(ω), u∞ (ω)i¯ > ²
2
¾¶
¯
¯
²
¯
¯
00
ω ∈ Ω : ¯hh(ω), Sn (ω)i − hh(ω), u∞ (ω)i¯ > ²
≤ µ(Ω\Ap ) + < ² .
2
Thus hh, Sn00 i → hh, u∞ i in measure.
2) Finally, if (wn )n is any other w-tight sequence, with wn ∈ co{u0i : i ≥ n},
then the same method applied to (vn )n gives for (wn )n the existence of a subsequence (wn0 )n of (wn )n and of a function v∞ ∈ L1X such that
(u0n )n and (wn0 )n
∀ h ∈ L0X ∗ [X] ,
(hh, wn0 i)n
σ(X, D)-K-converge a.e. to v∞
K-converges in measure to hh, v∞ i .
Since (u0n )n σ(X, D)-K-converges a.e. to u∞ , we get v∞ = u∞ a.e., so the proof
is complete.
Now we give the version of Theorem 1 relative to the s-tightness assumption.
Theorem 2. Let (un )n be a bounded sequence in L1X such that the following
tightness condition holds
T2 : Given any subsequence (uki )i of (un )n , there exists a s-tight sequence
(vn )n with vn ∈ co{uki : i ≥ n}.
Let D be a countable σ(X ∗ , X)-dense subset of the unit ball, B ∗ , of X ∗ . Then
there exist a function u∞ in L1X and a subsequence (u0n )n of (un )n such that
(a) (u0n )n σ(X, D)-K-converges a.e. to u∞ ;
A NEW EXTENSION OF KOMLÓS’ THEOREM
123
(b2 ) Every s-tight sequence (vn )n , with vn ∈ co{u0i : i ≥ n} has a subsequence
(vn0 )n such that
(vn0 )n
σ(X, D)-K-converges a.e. to u∞ ,
(vn0 )n
K-converges in measure to u∞ .
Proof: The proof is almost the same as the one given above. We will only
consider the points which need to be modified. Since each multifunction Kq has
0 ) s-converges a.e. to u∞ . Therefore, by
s-compact values, the sequence (Sn,q
n
q
Lebesgue–Vitali’s theorem,
∀ p, ∀ q ,
Z
Ap
0
|Sn,q
− u∞
q | dµ → 0 .
Consequently, by virtue of (11), for each p the sequence (1Ap Sn0 )n is relatively
strongly compact in L1X . Hence, (Sn0 )n is precompact in measure. If u∞ is a limit
point of (Sn0 )n with respect to the topology of convergence in measure, then by
(4),
∀ x∗ ∈ D , βx∗ (ω) = hx∗ , u∞ (ω)i a.e.
Therefore (Sn0 )n converges in measure to u∞ . The remainder of the proof follows
easily from the proof of Theorem 1.
Every w-tight (resp. s-tight) sequence satisfies T1 (resp. T2 ). So we have the
following
Corollary 3. Let (un )n be a bounded sequence in L1X and D a countable
σ(X ∗ , X)-dense subset of B ∗ . If (un )n is w-tight (resp. s-tight), then there exist
a function u∞ in L1X and a subsequence (u0n )n of (un )n for which (a) and (b01 )
(resp. (a) and (b02 )) hold, where
(b01 ) ∀ h ∈ L0X ∗ [X], (hh, u0n i)n K-converge in measure to hh, u∞ i;
(b02 ) (u0n )n K-converges in measure to u∞ .
The following result is due to E.J. Balder [[5], Th. B].
Corollary 4. Assume that X is not necessarily separable. Let (un )n be a
bounded sequence in L1X such that {un (ω) : n ∈ N} is a.e. relatively w-compact.
Then there exist a subsequence of (un )n that w-K-converges a.e. to a Bochner
integrable function.
124
M. SAADOUNE
Proof: By the Pettis measurability theorem, we may assume that X is
separable. So by Corollary 3, there exist a subsequence of (un )n and a function
in L1X for which (a) holds. Since {un (ω) : n ∈ N} is a.e. relatively w-compact, we
conclude, by Krein and Eberlein–Šmulian’s theorem, that the convergence in (a)
extends to all x∗ ∈ X ∗ .
Corollary 5. Assume that X ∗ is strongly separable. Let (un )n be a bounded
sequence in L1X such that the condition T1 holds. Then there exist a subsequence
of (un )n that w-K-converges a.e. to a Bochner integrable function.
Proof: Let D be a countable strongly dense subset of B ∗ . Let (u0n )n and
u∞ be as obtained in Theorem 1. By Komlós’ theorem and by extracting a
subsequence if necessary, we may suppose that the sequence (|u0n |)n K-converges
a.e. to an integrable function from Ω into R+ . Hence, for every further subseP
quence (u0ni )i of (u0n )n , the sequence ( n1 ni=1 u0ni (ω))n is a.e. bounded. Therefore,
since (u0n )n σ(X, D)-K-converges a.e. to u∞ and D is dense, the sequence (u0n )n
w-K-converges a.e. to u∞ .
From Corollary 5 we deduce the following result, which is due to E.J. Balder
[[5], Th. A].
Corollary 6. Assume that X is a reflexive Banach space (not necessarily separable). Let (un )n be a bounded sequence in L1X . Then there exist a
subsequence of (un )n that w-K-converges a.e. to a Bochner integrable function.
Proof: By the Pettis measurability theorem, we may assume that X is
separable. Furthermore, since X is reflexive, X ∗ is strongly separable. And, by
the boundedness assumption and reflexivity of X, the sequence (un )n is w-tight.
Thus we return to the situation of Corollary 5.
4 – Applications
As an immediate application of our results we have the following theorem,
which is due to C. Castaing [[11], Th. 4.1, p. 2. 14], see also [[1], Th. 6].
Theorem 7. Let H be a w-tight uniformly integrable subset of L1X . Then H
is sequentially relatively weakly compact (and hence relatively weakly compact)
in L1X .
A NEW EXTENSION OF KOMLÓS’ THEOREM
125
Proof: Let (un )n be a sequence in H. Let (u0n )n and u∞ be as obtained in
Corollary 3. By UI and (b01 ), it follows from Lebesgue–Vitali’s theorem that
∀ h ∈ L∞
X ∗ [X] ,
³Z
Ω
hh, u0n i dµ
´
n
K-converges to
Z
Ω
hh, u∞ i dµ .
Hence, by using the remark given in section 2, we get
u0n → u∞
weakly in L1X .
Thus H is sequentially relatively weakly compact. Then, by Eberlein–Šmulian’s
theorem [[22], Th. 18, A], H is relatively weakly compact in L1X .
Now we give a very general criterion for relative weak compactness in L 1X .
Theorem 8. Let H be a bounded subset of L1X . Then the following statements are equivalent:
(i) H is relatively weakly compact in L1X .
(ii) H is UI and every sequence in H satisfies T1 .
(iii) H is UI and every sequence in H satisfies T2 .
Proof: Suppose (i) holds. Then H is UI (see [16], [9]). Let (un )n be a
sequence in H. There exists a subsequence (u0n )n of (un )n that converges weakly
to some u ∈ L1X . Then, by Mazur’s theorem, there exists a sequence (vn )n , with
vn ∈ co{u0i : i ≥ n} such that |vn − u|L1 → 0. Hence, there exists a subsequence
X
(vn0 )n of (vn )n that converges a.e. to u. Therefore (vn0 )n is s-tight. This proves
(iii). The implication (iii)⇒(ii) is obvious. Now let us prove the main implication
of the theorem (ii)⇒(i). Suppose (ii) holds. By Eberlein–Šmulian’s theorem, it
is enough to show that H is sequentially relatively weakly compact. Let (un )n be
a sequence in H. Since (un )n is bounded and satisfies T1 , by Theorem 1, there
exist a subsequence (u0n )n of (un )n and a function u∞ ∈ L1X for which (b1 ) holds.
0
0
For a fixed h in L∞
X ∗ [X], choose a subsequence (uki )i of (un )n such that
lim
Z
i→+∞ Ω
hh, u0ki i dµ = lim sup
n
Z
Ω
hh, u0n i dµ .
By T1 , there exists a w-tight sequence (vn )n , with vn ∈ co{u0ki : i ≥ n} ⊂ co{u0i :
i ≥ n}. So, by (b1 ), every subsequence of (vn )n has a further subsequence (vn0 )n
such that
(hh, vn0 i)n K-converges in measure to hh, u∞ i
126
M. SAADOUNE
then
³Z
Ω
hh, vn0 i dµ
´
K-converges to
n
Z
Ω
hh, u∞ i dµ .
This is equivalent to
Z
Ω
hh, vn0 i dµ
→
Z
Ω
Z
Ω
hh, u∞ i dµ .
Consequently,
Z
Ω
hh, vn i dµ →
hh, u∞ i dµ .
Since each vn is of the form
vn =
X
λi u0ki+n ,
with λi ≥ 0 and
i≤kn
we have
Z
Ω
X
λi = 1 ,
i≤kn
hh, u∞ i dµ = lim
n→+∞
= lim
X
i≤kn
Z
i→+∞ Ω
Then
lim sup
n
Z
λi
Ω
hh, u0ki+n i dµ
hh, u0ki i dµ .
hh, u0n i dµ =
Z
hh, u0n i dµ
Z
Ω
Z
Ω
hh, u∞ i dµ .
Similarly, we obtain
lim inf
n
Hence
lim
Z
Z
Ω
n→+∞ Ω
=
hh, u0n i dµ =
Z
Ω
Ω
hh, u∞ i dµ .
hh, u∞ i dµ .
This proves (i).
Remark 1. Theorem 8 allows to recover a recent criterion for relative
weak compactness, which is due to A. Ülger [30] and J. Diestel–W.M. Ruess–
W. Schachermayer [17] (by the Pettis measurability theorem, the hypothesis “X
is separable” can be relaxed). In [17] the proof is based on the characterization of
the weak compactness by the iterated limit condition (see [[22], Lemma 1, 19 A]).
Remark 2. Considering constant functions in X, we recover also A. Ülger’s
lemma [[30], Lemma 2.1]. Recall that this author proved his result by using
James’s characterization of weak compactness via norm-attaining functionals.
A NEW EXTENSION OF KOMLÓS’ THEOREM
127
ACKNOWLEDGEMENT – The author thanks C. Castaing, M. Valadier and L. Thibault
for helpful comments.
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Mohammed Saadoune,
Département de Mathématiques, Université Ibnou Zohr,
Lot. Addakhla B.P. 28-S, Agadir – MAROC