2396 CHAPTER 70. MEASURABILITY WITHOUT UNIQUENESS

Definition 70.2.4 Let Γ(ω)≡ ∩∞n=1Γn (ω).

Lemma 70.2.5 Γ is a nonempty F measurable set valued function having values in thecompact sub-sets of X. There exists a measurable selection γ . For γ a F measurableselection, (t,ω)→ γ (t,ω) is P measurable. Also, for each ω, there exists a subsequence,un(ω) (·,ω) such that for each k,

γk (t,ω) = limn(ω)→∞

f(un(ω) (t,ω)

)k = lim

n(ω)→∞

mk

∫ t

lmk (t)

⟨φ rk

,un(ω) (s,ω)⟩

V,V ′ds

Proof: Consider Γ(ω) =∩∞n=1Γn (ω) . Then ω→ Γ(ω) is a compact set valued map in

X . It is nonempty because each Γn (ω) is nonempty and compact, and these sets are nested.Is it F measurable? Each Γn is compact valued and F measurable. Hence if F is closed,

Γ(ω)∩F = ∩∞n=1Γ

n (ω)∩F

and the left is non empty if and only if each Γn (ω)∩F ̸= /0. Hence 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 Theorem 70.1.2, [10], [70]. SinceΓn (ω) is compact, the measurability of Γn, that Γn− (U)∈F for U open implies the strongmeasurability of Γn, that Γn− (F) ∈F . Thus ω → Γ(ω) is non empty compact valued inX and F measurable.

From standard theory of measurable multi-functions, Theorem 70.1.2, [10], [70], thereexists a F measurable selection ω→ γ (ω) with γ (ω) ∈ Γ(ω) for each ω . Now it followsthat t → γk (t,ω) is continuous. This is what it means for γ (ω) ∈ X . What of the productmeasurability of γk? We know that ω → γk (ω) is F measurable into C ([0,T ]) and sosince pointwise evaluation is continuous, ω → γk (t,ω) is F measurable. Then since t →γk (t,ω) is continuous, it follows that γk is a P measurable real valued function and that γ

is a P measurable R∞ valued function.Since γ (ω) ∈ Γ(ω) , it follows that for each n,γ (ω) ∈ Γn (ω) . Therefore, there exists

jn ≥ n such that for each ω,

d (f(u jn (·,ω)) ,γ (ω))< 2−n

It follows that, taking a suitable subsequence, denoted as{

un(ω) (·,ω)}

,

γ (ω) = limn(ω)→∞

f(un(ω) (·,ω)

)for each ω . In particular, for each k

γk (t,ω) = limn(ω)→∞

f(un(ω) (t,ω)

)k = lim

n(ω)→∞

mk

∫ t

lmk (t)

⟨φ rk

,un(ω) (s,ω)⟩

V,V ′ds (70.2.2)