77.2. SOME FUNDAMENTAL THEOREMS 2597
Thus u→ Bu(0) is continuous into V ′. If p < 2, then you do something similar.(∫ T
0∥Bu(0)−Bv(0)∥p
V ′ dt)1/p
≤(∫ T
0∥Bu(t)−Bv(t)∥p
V ′ dt)1/p
+
(∫ T
0
∥∥∥∥∫ t
0(Bu)′ (s)− (Bv)′ (s)ds
∥∥∥∥p
dt)1/p
∥Bu(0)−Bv(0)∥V ′ T1/p ≤ ∥B∥∥u− v∥Lp +C (T )
∥∥(Bu)′− (Bv)′∥∥
Lp′ ([0,T ];V ′)
≤ C (∥B∥ ,T )∥u− v∥X .
However, one could just as easily have done this for an arbitrary s < T by repeating theargument for
Bu(t) = Bu(s)+∫ t
s(Bu)′ (r)dr
Thus this mapping is certainly continuous into V ′. The last assertion is similar.Also of use will be the following generalization of the Ascoli Arzela theorem. [117],
Theorem 69.5.4.
Theorem 77.2.3 Let q > 1 and let E ⊆W ⊆ X where the injection map is continuous fromW to X and compact from E to W. Let S be defined by{
u such that ||u(t)||E ≤ R for all t ∈ [a,b] , and ∥u(s)−u(t)∥X ≤ R |t− s|1/q}.
Thus S is bounded in L∞ (0,T,E) and in addition, the functions are uniformly Holder con-tinuous into X . Then S ⊆ C ([a,b] ;W ) and if {un} ⊆ S, there exists a subsequence,
{unk
}which converges to a function u ∈C ([a,b] ;W ) in the following way.
limk→∞
∣∣∣∣unk −u∣∣∣∣
∞,W = 0.
Next is a major measurable selection theorem which forms an essential part of showingthe existence of measurable solutions. See Theorem 70.2.1. The following is not dependenton there being a measure but in the applications there is typically a probability measure andoften a set of measure zero which occurs in a natural way so an exceptional set of measurezero is included in the statement of the theorem but it has absolutely nothing to do with aset of measure zero as will be seen by just letting the exceptional set be /0.
Theorem 77.2.4 Let V be a reflexive separable Banach space with dual V ′, and let p, p′
be such that p > 1 and 1p + 1
p′ = 1. Let the functions t → un (t,ω), for n ∈ N, be in
Lp′ ([0,T ] ;V ′) and (t,ω)→ un (t,ω) be B ([0,T ])×F ≡P measurable into V ′. Sup-pose there is a set of measure zero N ⊆Ω such that if ω /∈ N, then
supt∈[0,T ]
∥un (t,ω)∥V ′ ≤C (ω) ,