77.2. SOME FUNDAMENTAL THEOREMS 2599

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

t→ fk (u j (·,ω))(t)}

j are uniformly boundedand equicontinuous because they are in C0,1 ([0,T ]). Indeed, we have for ω /∈ N,

∣∣ fk (u j (·,ω))(t)∣∣= ∣∣∣∣∣mk

∫ t

lmk (t)

⟨φ rk

,u j (s,ω)⟩

Vds

∣∣∣∣∣≤C (ω)∥∥∥φ rk

∥∥∥V

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∣∣C (ω)

∥∥∥φ rk

∥∥∥V ′.

By Lemma 77.2.5, 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.

Lemma 77.2.6 The mapping ω→ Γn (ω) is an F measurable set-valued map with valuesin X. If σ is a measurable selection, then for each t, ω → σ (t,ω) is F measurable and(t,ω)→ σ (t,ω) is B ([0,T ])×F measurable.

We note that if σ is a measurable selection then σ (ω) ∈ Γn (ω), so σ = σ (·,ω) is acontinuous function. To have σ measurable would mean that σ

−1k (open) ∈F , where the

open set is in C ([0,T ]).Proof: Let O be a basic open set in X . Then O = ∏

∞k=1 Ok, where Ok is a proper open

set of C ([0,T ]) only for k ∈ {k1, · · · ,kr}. Thus there is a proper open set in these positionsand in every other position the open set is the whole space C ([0,T ]) .We need to show that

Γn− (O)≡ {ω : Γ

n (ω)∩O ̸= /0} ∈F .

Now, Γn− (O) = ∩ri=1

{ω : Γn (ω)ki

∩Oki ̸= /0}

, so we consider whether{ω : Γ

n (ω)ki∩Oki ̸= /0

}∈F . (77.2.2)