320 15 A Compactness Principle
Note — and this is aLeitmotivof the present paper — that, for sequences
(xn)n≥ 1 in a vector space, passing to convexcombinations usually does not
cost more than passing to a subsequence. In most applications the main prob-
lem is to find a limitx 0 ∈X and typically it does not matter whether
x 0 = limkxnk for a subsequence (xnk)k≥ 1 orx 0 = limnynfor a sequence
of convex combinationsyn∈conv{xn,xn+1,...}.
If one passes to the case of non-reflexive Banach spaces there is — in gen-
eral — no analogue to Theorem 15.1.2 pertaining to any bounded sequence
(xn)n≥ 1 , the main obstacle being that the unit ball fails to be weakly com-
pact. But sometimes there are Hausdorff topologies on the unit ball of a
(non-reflexive) Banach space which have some kind of compactness proper-
ties. A noteworthy example is the Banach spaceL^1 (Ω,F,P) and the topology
of convergence in measure.
Theorem 15.1.3.Given a bounded sequence(fn)n≥ 1 ∈ L^1 (Ω,F,P)then
there are convex combinations
gn∈conv{fn,fn+1,...)}such that(gn)n≥ 1 converges in measure to someg 0 ∈L^1 (Ω,F,P).
The preceding theorem is a somewhat vulgar version of Komlos’ theo-
rem [K 67]. Note that Komlos’ result is more subtle as it replaces the convex
combinations (gn)n≥ 1 by the Cesaro-means of a properly chosen subsequence
(fnk)k≥ 1 of (fn)n≥ 1.
But the abovevulgar versionof Komlos’ theorem has the advantage that
it extends to the case ofL^1 (Ω,F,P;E) for reflexive Banach spacesEas we
shall presently see (Theorem 15.1.4 below), while Komlos’ theorem does not.
(J. Bourgain [B 79] proved that the precise necessary and sufficient condition
for the Komlos theorem to hold forE-valued functions is thatL^2 (Ω,F,P;E)
has the Banach-Saks property; compare [G 79] and [S 81].)
Here is the vector-valued version of Theorem 15.1.3:
Theorem 15.1.4.IfE is a reflexive Banach space and(fn)n≥ 1 abounded
sequence inL^1 (Ω,F,P;E), we may find convex combinations
gn∈conv{fn,fn+1,...}andg 0 ∈L^1 (Ω,F,P;E)such that(gn)n≥ 1 converges tof 0 almost surely, i.e.,
lim
n→∞
‖gn(ω)−g 0 (ω)‖E=0 for a.e.ω∈Ω.The preceding theorem seems to be of folklore type and to be known to
specialists for a long time (compare also [DRS 93]). We shall give a proof in
Sect. 15.2 below.
Let us have a closer look at what is really happening in Theorems 15.1.3
and 15.1.4 above by following the lines of KadeˇcandPelczy ́nski [KP 65].