78.4. PROGRESSIVELY MEASURABLE SOLUTIONS 2677

where the notation means qτr (t) ≡ q(t ∧ τr). Then, since qτr is uniformly bounded, all ofthe necessary estimates and measurability for the solution to the above corollary hold forAr replacing A. Therefore, there exists a solution wr to the inclusion

(Bwr)′ (·,ω)+Ar (wr (·,ω) ,ω) ∋ f (·,ω) , Bwr (0,ω) = Bu0 (ω) , t ∈ [0,T −σ/2]

Now for fixed ω,qτr (t,ω) does not change for all r large enough. This is because it is acontinuous function of t and so is bounded on the interval [0,T −σ/2]. Thus, for r largeenough and fixed ω , qτr (t,ω) = q(t,ω) . Thus, we obtain

⟨Bwr (t,ω) ,wr (t,ω)⟩+∫ t

0∥wr (s,ω)∥p

V ds≤C (ω) (78.3.31)

Now, as before in the proof of Theorem 78.3.2 one can pass to a limit involving a subse-quence, as r (ω)→ ∞ and obtain a solution to the integral equation

Bw(t,ω)−Bu0 (ω)+∫ t

0u∗ (s,ω)ds =

∫ t

0f (s,ω)ds, t ∈ [0,T −σ ]

where u∗ (ω) ∈ A(w(s,ω)+q(s,ω) ,ω) and u∗,w are measurable into V ′[0,T−σ ]. Now letu(t,ω) = w(t,ω)+q(t,ω) .

The last claim follows from letting t = a in the top equation and then subtracting thisfrom the top equation with t > a.

78.4 Progressively Measurable SolutionsIn the context of uniqueness of the evolution initial value problem for fixed ω, one canprove theorems about progressively measurable solutions fairly easily.

First is a definition of the term progressively measurable.

Definition 78.4.1 Let Ft be an increasing in t set of σ algebras of sets of Ω where (Ω,F )is a measurable space. Thus each Ft is a σ algebra and if s≤ t, then Fs ≤Ft . This set ofσ algebras is called a filtration. A set S ⊆ [0,T ]×Ω is called progressively measurable iffor every t ∈ [0,T ] ,

S∩ [0, t]×Ω ∈B ([0, t])×Ft

Denote by P the progressively measurable sets. This is a σ algebra of subsets of [0,T ]×Ω.A function g is progressively measurable if X[0,t]g is B ([0, t])×Ft measurable for each t.

Let A satisfy the conditions 78.3 - 78.3 but the last condition will be modified as follows.

Condition 78.4.2 For each t ≤ T, if ω → u(·,ω) is Ft measurable into V[0,t], then thereexists a Ft measurable selection of A

(X[0,t]u(·,ω) ,ω

)into V ′[0,t].

Note that u(·,ω) is in V[0,t] so u(t,ω) ∈V .The theorem to be shown is the following.