2600 CHAPTER 77. STOCHASTIC INCLUSIONS

From the definition of Γn (ω) , this is equivalent to the condition that

fki (XNC (ω)u j (·,ω)) = (f(XNC (ω)u j (·,ω)))ki∈ Oki

for some j ≥ n, and so the set in 77.2.2 is of the form

∪∞j=n

{ω : (f(XNC (ω)u j (·,ω)))ki

∈ Oki

}.

Now ω → (f(XNC (ω)u j (·,ω)))kiis F measurable into C ([0,T ]) and so the above set is

in F . To see this, let g ∈C ([0,T ]) and consider the inverse image of the ball with radius rand center g,

B(g,r) ={

ω :∥∥∥(XNC (ω) f(u j (·,ω)))ki

−g∥∥∥

C([0,T ])< r}.

By continuity considerations,∥∥∥(XNC (ω) f(u j (·,ω)))ki−g∥∥∥

C([0,T ])

= supt∈Q∩[0,T ]

∣∣∣(XNC (ω) f(u j (t,ω)))ki−g(t)

∣∣∣ ,which is the sup over countably many F measurable functions. Thus, it is F measurable.Since every open set is the countable union of such balls, it follows that the claim about Fmeasurability is valid. Hence, Γn− (O) is F measurable whenever O is a basic open set.

Now, X is a separable metric space and so every open set is a countable union of thesebasic sets. Let U ⊆ X be open with U = ∪∞

l=1Ol where Ol is a basic open set as above.Then,

Γn− (U) = ∪∞

l=1Γn− (Ol) ∈F .

The existence of a measurable selection follows from the standard theory of measurablemulti-functions [10, 70] see [70] starting on Page 141 for all the necessay stuff on measur-able multifunctions or Section 48. If σ is one of these measurable selections, the evalua-tion at t is F measurable. Thus, ω → σ (t,ω) is F measurable with values in R∞. Also,t→ σ (t,ω) is continuous, and so it follows that in fact σ is product measurable as claimed.

Definition 77.2.7 Let Γ(ω)≡ ∩∞n=1Γn (ω).

Lemma 77.2.8 Γ is a nonempty F measurable set-valued function with values in com-pact subsets of X. There exists a measurable selection γ such that (t,ω)→ γ (t,ω) is Pmeasurable. Also, for each ω, there exists a subsequence, un(ω) (·,ω) such that for each k,

γk (t,ω) = limn(ω)→∞

f(XNC (ω)un(ω) (t,ω)

)k

= limn(ω)→∞

mk

∫ t

lmk (t)

⟨φ rk

,XNC (ω)un(ω) (s,ω)⟩

Vds.