2604 CHAPTER 77. STOCHASTIC INCLUSIONS

Proof: This follows right away from Tychonoff’s theorem and the compactness of theembedding of the Holder space into C ([0,1]).

Now, for m∈N and φ ∈V ′, define lm(t)≡max(0, t− (1/m)) and ψm,φ : V →C ([0,T ])by

ψm,φ u(t)≡∫ T

0

⟨mφX[lm(t),t] (s) ,u(s)

⟩V ds = m

∫ t

lm(t)⟨φ ,u(s)⟩V ds.

Here, X[lm(t),t] (·) is the characteristic function of the interval [lm(t), t] and ⟨·, ·⟩V = ⟨·, ·⟩Vis the duality pairing between V and V ′.

Let D = {φ r}∞

r=1 denote a countable subset of V ′. Then the pairs (φ ,m) for φ ∈D and

m ∈ N form a countable set. Let(

mk,φ rk

)denote an enumeration of the pairs (m,φ) ∈

N×D . To simplify the notation, we set

fk (u)(t)≡ ψmk,φ rk(u)(t) = mk

∫ t

lmk (t)

⟨φ rk

,u(s)⟩

Vds.

For fixed ω /∈ N and k, the functions{

t→ fk (u j (·,ω))(t)}

j are uniformly bounded

and equicontinuous because they are in C0,1/p′ ([0,T ]). Indeed, we have for ω /∈ N,

∣∣ fk (u j (·,ω))(t)∣∣ =

∣∣∣∣∣mk

∫ t

lmk (t)

⟨φ rk

,u j (s,ω)⟩

Vds

∣∣∣∣∣≤ m

∥∥∥φ rk

∥∥∥T 1/p′∣∣∣∣∫ T

0

∥∥u j (s,ω)∥∥p

V ds∣∣∣∣1/p

≤C (ω)m∥∥∥φ rk

∥∥∥V ′

T 1/p′

and for t ≤ t ′ ∣∣ fk (u j (·,ω))(t)− fk (u j (·,ω))(t ′)∣∣

≤

∣∣∣∣∣mk

∫ t

lmk (t)

⟨φ rk

,u j (s,ω)⟩

Vds−mk

∫ t ′

lmk (t′)

⟨φ rk

,u j (s,ω)⟩

Vds

∣∣∣∣∣≤ 2mk

∣∣t ′− t∣∣1/p′C (ω)

∥∥∥φ rk

∥∥∥V ′.

By Lemma 77.2.11, the set of functions{XNC (ω) f(u j (·,ω))

}∞

j=n is pre-compact in X =

∏k C ([0,T ]) . We now define a set valued map Γn : Ω→ X by

Γn (ω)≡ ∪ j≥n

{XNC (ω) f(u j (·,ω))

},

where the closure is taken in X . Then Γn (ω) is the closure of a pre-compact set in X andso Γn (ω) is compact in X . From the definition, a function f is in Γn (ω) if and only ifd (f,XNC (ω) f(wl))→ 0 as l→ ∞, where each wl is one of the u j (·,ω) for j ≥ n. In thetopology on X , this happens iff for every k,

fk (t) = liml→∞

mk

∫ t

lmk (t)

⟨φ rk

,XNC (ω)wl (s,ω)⟩

Vds,

where the limit is the uniform limit in t.