24.12. CONDITIONAL EXPECTATION IN BANACH SPACES 701
if k is chosen large enough. Now consider the first in 24.52. By Jensen’s inequality, Lemma10.15.1,
k
∑i=1
∫ ti
ti−1
ε
∥∥∥∥ 1ti− ti−1
∫ ti
ti−1
(un (t)−un (s))ds∥∥∥∥p
Edt
≤k
∑i=1
∫ ti
ti−1
ε1
ti− ti−1
∫ ti
ti−1
∥un (t)−un (s)∥pE dsdt
≤ ε2p−1k
∑i=1
1ti− ti−1
∫ ti
ti−1
∫ ti
ti−1
(∥un (t)∥p +∥un (s)∥p)dsdt
= 2ε2p−1k
∑i=1
∫ ti
ti−1
(∥un (t)∥p)dt = ε (2)(2p−1)∥un∥Lp([a,b],E) ≤Mε
Now pick ε sufficiently small that Mε < η p
10p and then k large enough that the second termin 24.52 is also less than η p/10p. Then it will follow that
∥ūn−un∥Lp([a,b],W ) <
(2η p
10p
)1/p
= 21/p η
10≤ η
5
Thus k is fixed and un at a step function with k steps having values in E. Now usecompactness of the embedding of E into W to obtain a subsequence such that {un} isCauchy in Lp ([a,b] ;W ) and use this to contradict 24.51. The details follow.
Suppose un (t) = ∑ki=1 un
i X[ti−1,ti) (t) . Thus ∥un (t)∥E = ∑ki=1 ∥un
i ∥E X[ti−1,ti) (t) and soR≥
∫ ba ∥un (t)∥p
E dt = Tk ∑
ki=1 ∥un
i ∥pE Therefore, the {un
i } are all bounded. It follows that af-ter taking subsequences k times there exists a subsequence
{unk
}such that unk is a Cauchy
sequence in Lp ([a,b] ;W ) . You simply get a subsequence such that unki is a Cauchy se-
quence in W for each i. Then denoting this subsequence by n,
∥un−um∥Lp(a,b;W ) ≤ ∥un−un∥Lp(a,b;W )+∥un−um∥Lp(a,b;W )+∥um−um∥Lp(a,b;W )
≤ η
4+∥un−um∥Lp(a,b;W )+
η
4< η
provided m,n are large enough, contradicting 24.51. ■
24.12 Conditional Expectation in Banach SpacesLet (Ω,F ,P) be a probability space and let X ∈ L1 (Ω;R). Also let G ⊆F where G isalso a σ algebra. Then the usual conditional expectation is defined by∫
AXdP =
∫A
E (X |G )dP
where E (X |G ) is G measurable and A ∈ G is arbitrary. Recall this is an application of theRadon Nikodym theorem. Also recall E (X |G ) is unique up to a set of measure zero.
I want to do something like this here. Denote by L1 (Ω;E,G ) those functions inL1 (Ω;E) which are measurable with respect to G .