328 15 A Compactness Principle
E[fnl∧βl(g+1)−fnl∧βl(g+1)∧K(g+1)]
=E[fnl−fnl∧K(g+1)]−E[fnl−fnl∧βl(g+1)]≤δ(K)−(
δ(∞)−δ(∞)
2 k(K))
≤δ(∞)−δ(K)+
δ(∞)
2 k(K).
The latter expression clearly tends to 0 asK→∞.
Corollary 15.2.2.If the sequenceβkis such thatfnk∧βk(g+1)is uniformly
integrable, then there also exists a sequenceγksuch thatγβkktends to infinity
and such that the sequencefnk∧γk(g+1)remains uniformly integrable.
Proof.In order to show the existence ofγkwe proceed as follows. The sequence
hk=βk(g+1) (^1) {fnk≥βk(g+1)}
tends to zero inL^1 (P), since the sequencefnk∧βk(g+1) is uniformly integrable
andP[fnk≥βk(g+1)]≤β^1 k→0. Let nowαkbe a sequence that tends to
infinity but so thatαkhkstill tends to 0 inL^1 (P). If we defineγk=αkβkwe
have that
fnk∧γk(g+1)≤fnk∧βk(g+1)+αkhk
and hence we obtain the uniform integrability offnk∧γk(g+1).
Remark 15.2.3.In most applications of the Kadeˇc-Pelczy ́nski decomposition
theorem, we can takeg= 0. However, in Sect. 15.4, we will need the easy
generalisation to the case wheregis a non-zero integrable non-negative func-
tion. The general case can in fact be reduced to the caseg=0byreplacing
the functionsfnby(gf+1)n and by replacing the measurePby the probability
measureQdefined asdQ=E(g[g+1)+1]dP.
Remark 15.2.4.We will in many cases drop indices likenkand simply suppose
that the original sequence (fn)n≥ 1 already satisfies the conclusions of the
theorem. In most cases such passing to a subsequence is allowed and we will
abuse this simplification as many times as possible.
Remark 15.2.5.The sequence of sets{fn >βn(g+1)} is, of course, not
necessarily a disjoint sequence. In case we need two by two disjoint sets
we proceed as follows. By selecting a subsequence we may suppose that∑
n>kP[fn >βn(g+1)]≤ εk, where the sequence of strictly positive
numbers (εk)k≥ 1 is chosen in such a way that
∫
BfkdP<^2−kwheneverP[B]<εk. It is now easily seen that the sequence of sets (An)n≥ 1 defined by
An={fn>βn(g+1)}\
⋃
k>n{fk>βk(g+1)}will do the job.