77.5. THE MAIN RESULT 2625

Note that we are not assuming that ω → A(u,ω) is measurable, only that it has a mea-surable selection and of course the upper semicontinuity and that the values are closed andconvex. Also note that ω → Γ(ω) is measurable so there is a dense subset of measurablefunctions {zk (ω)} each being measurable into X . However, we don’t know much aboutΓ(ω) other than it is measurable into X .

Also V could be replaced with Lp (I,V ) where I is any interval and nothing changes.The condition leading to 77.5.26 will typically be satisfied. For example, suppose u∗ ∈

A(u,ω) means that

u∗ (t) ∈ A(

t,u(t) ,∫ t

0u(s)ds,ω

)for a.e. t. where A has values in P (V ′). Then to say that u∗X[0,T̂ ] ∈ A

(uX[0,T̂ ],ω

)for

each T̂ in an increasing sequence converging to T would imply the above holding for a.e. t.While the above is the typical situation one would expect to see, the following propositionis also interesting.

Proposition 77.5.4 Suppose A(·,ω) : V →P (V ′) is upper semicontinuous as a map fromthe strong topology of V to the weak topology of V ′ and has closed convex values. Then if

u∗X[0,T̂ ] ∈ A(

uX[0,T̂ ],ω)

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

u∗ ∈ A(u,ω) (77.5.36)

Proof: Let T̂n ↑ T such that u∗X[0,T̂n] ∈ A(

uX[0,T̂n],ω). Then if u∗ /∈ A(u,ω) , there

exists z ∈ V such that⟨u∗,z⟩> r > ⟨w∗,z⟩

for all w∗ ∈ A(u,ω). Now u∗X[0,T̂n]→ u∗ in V ′ and uX[0,T̂n]→ u in V . Letting O be theweakly open set, {z∗ : ⟨z∗,z⟩< r} , it follows that this O is a weakly open set which containsA(u,ω). Hence, by upper semicontinuity,

⟨u∗X[0,T̂n],z

⟩< r for all n large enough. Hence,

passing to a limit, one obtains ⟨u∗,z⟩> r ≥ ⟨u∗,z⟩ , a contradiction. Thus u∗ ∈ A(u,ω).Let r > max(p̂,2). Recall that p̂ ≥ p and the growth had to do with p̂. Let U and UI

be defined by analogy with V and VI where U ≡ Lr ([0,T ] ,U). Here U is a Hilbert spacewhich is dense in V and embedds compactly into V,∥u∥U ≥ ∥u∥V . Also let F : U →U ′ bethe duality map for r. Thus

∥Fu∥U ′ = ∥u∥r−1U , ⟨Fu,u⟩= ∥u∥r

U

Also define the following notation for small positive h.

τhg(t)≡{

g(t−h) if t > h0 if t ≤ h