2606 CHAPTER 77. STOCHASTIC INCLUSIONS

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 necessary stuff on mea-surable multifunctions or Section 48. If σ is one of these measurable selections, the evalu-ation 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 77.2.13 Let Γ(ω)≡ ∩∞n=1Γn (ω).

Lemma 77.2.14 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.

Proof: From the definition of Γ(ω) = ∩∞n=1Γn (ω) it follows that ω → Γ(ω) is a com-

pact set-valued map in X and is nonempty because each Γn (ω) is nonempty and compact,and the Γn (ω) are nested. We next show that ω → Γ(ω) is F measurable. Indeed, eachΓn is compact valued and F measurable so, if F is closed,

Γ(ω)∩F = ∩∞n=1Γ

n (ω)∩F,

and the left-hand side is not empty if and only if each Γn (ω)∩F ̸= /0. Thus, for F closed,

{ω : Γ(ω)∩F ̸= /0}= ∩n {ω : Γn (ω)∩F ̸= /0} ,

and soΓ− (F) = ∩nΓ

n− (F) ∈F .

The last claim follows from the theory of multi-functions, see, e.g., [10, 70] or Section 48.The fact that Γn (ω) is compact, Γn is measurable and Γn− (U) ∈F , for U open, implythe strong measurability of Γn [10, 70] see also Section 48, and also that Γn− (F) ∈ F .Thus, ω → Γ(ω) is nonempty compact valued in X and F measurable. We are using thetheorem which says that when Γ has compact values, then one can conclude that strongmeasurability and measurability coincide. See Proposition 48.1.4. This is why we can saythat Γn− (F) ∈F .

The standard theory [10, 70], Section 48, also guarantees the existence of an F mea-surable selection ω → γ (ω) with γ (ω) ∈ Γ(ω), for each ω , and also that t → γk (t,ω)(the kth component of γ) is continuous. Next, we consider the product measurability of γk.We know that ω → γk (ω) is F measurable into C ([0,T ]) and since pointwise evaluationis continuous, ω → γk (t,ω) is F measurable. (This is nothing more than a case of thegeneral result that a continuous function of a measurable function is measurable.) Then,since t→ γk (t,ω) is continuous, it follows that γk is a P measurable real valued function