348 15 A Compactness Principle
as well as the negative parts of the super-martingales ((Hn·S)t)t∈R+areP-
uniformly integrable. Identifying the closed interval [0,∞] with the closed
interval [0,1], and identifying the processesSandHn·Swith process which
remain constant after timet= 1, we deduce from Theorem 15.D that we may
findKn∈conv{Hn,Hn+1,...},aw-admissible integrandH^0 and a process
(Vt)t∈R+such that
lims↘t
s∈Q+lim
n→∞
(Kn·S)s=Vt, a.s. fort∈R+and
lim
n→∞
(Kn·S)∞=V∞,
((H^0 ·S)t−Vt)t∈R+∪{∞} is increasing.
In particular ((Kn·S)∞)n≥ 1 converges almost surely to the random vari-
ableU∞which is dominated by (H^0 ·S)∞.
As (fn)n≥ 1 was assumed to converge in measure tof 0 we deduce that
f 0 ≤(H^0 ·S)∞, i.e.f 0 ∈C 1 ,w.
To pave the way for the proof of Theorem 15.D we start with some lemmas.Lemma 15.4.12.Under the assumptions of Theorem 15.D there is a sequence
of convex combinations
Kn∈conv{Hn,Hn+1,...},andasequence(Ln)n≥ 1 ofw-admissible integrands and there are cadl ag super-
martingalesV =(Vt)t∈R+andW=(Wt)t∈R+withW−V increasing such
that
Vt= lims↘t
s∈Q+lim
n→∞
(Kn·M)s, fort∈R+,a.s.Wt= lims↘t
s∈Q+lim
n→∞
(Ln·M)s, fort∈R+,a.s.and such thatWsatisfies the following maximality condition: For any sequence
(L ̃n)n≥ 1 ofw-admissible integrands such that
W ̃t= lim
ss∈↘Qt
+lim
n→∞
(L ̃n·M)sandW ̃−W increasing we have that ̃W=W.
Proof.By Theorem 15.1.3 we may findKn∈conv{Hn,Hn+1,...}such that,
for everyt∈Q+, the sequence ((Kn·M)t)n≥ 1 converges a.s. to a random
variableV̂t.Aswis assumed to be integrable we obtain that the process
(V̂t)t∈Q+is a super-martingale and therefore its cadlag regularisation,