77.7. PROGRESSIVELY MEASURABLE SOLUTIONS 2639

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 bounded condition 77.5.25, the condition on subintervals 77.5.26, thespecific boundedness estimate 77.5.27, the specific coercivity estimate involving B and Ain 77.5.29, and the limit condition 77.5.30. In place of the condition on the existence of ameasurable selection 77.5.34, we will assume the following condition.

Condition 77.7.2 For each t ≤ T, if ω → u(·,ω) is Ft measurable into V[0,t], then thereexists a Ft measurable selection of A(u(·,ω) ,ω) into V ′[0,t].

Note that u(·,ω) is in V[0,t] so u(t,ω) ∈V .In this section, we assume that ω → B(ω) is F0 measurable into L (W,W ′). For

convenience, here are the conditions used on A.For the operator A(·,ω) . A(·,ω) : VI→ V ′I for each I a subinterval of [0,T ] , I =

[0, T̂]

andA(·,ω) : VI →P(V ′I) is bounded, (77.7.66)

If, for u ∈ V ,u∗X[0,T̂ ] ∈ A

(uX[0,T̂ ],ω

)for each T̂ in an increasing sequence converging to T, then

u∗ ∈ A(u,ω) (77.7.67)

For some p̂≥ p, assume the specific estimate

sup{∥u∗∥V ′I : u∗ ∈ A(u,ω)

}≤ a(ω)+b(ω)∥u∥p̂−1

VI(77.7.68)

where a(ω) ,b(ω) are nonnegative. Note that the growth could be quadratic in case p = 2.This really just says there is polynomial growth. Also assume the coercivity condition:

lim||u||V→∞

u∈Xr

inf{2⟨u∗,u⟩V ′,V + ⟨Bu,u⟩(T ) : u∗ ∈ A(u,ω)}||u||V

= ∞, (77.7.69)

or alternatively the following specific estimate valid for each t ≤ T and for some λ (ω)≥ 0,

inf(∫ t

0⟨u∗,u⟩+λ (ω)⟨Bu,u⟩dt : u∗ ∈ A(u,ω)

)≥ δ (ω)

∫ t

0∥u∥p

V ds−m(ω) (77.7.70)

where m(ω) is some nonnegative constant, δ (ω)> 0. Note that the estimate is a coercivitycondition on λB+A rather than on A but is more specific than 77.5.28.

Let U be a Banach space dense in V and that if ui ⇀ u in VI and u∗i ∈A(ui) with u∗i ⇀ u∗

in V ′I and (Bui)′⇀ (Bu)′ in U ′

rI , ⇀ denoting weak convergence, then if

lim supi→∞

⟨u∗i ,ui−u⟩V ′I ,VI≤ 0