Chapter 48

Multifunctions and Their Measurability48.1 The General Case

Let X be a separable complete metric space and let (Ω,C ,µ) be a set, a σ algebra ofsubsets of Ω, and a measure µ such that this is a complete σ finite measure space. Alsolet Γ : Ω→PF (X) , the closed subsets of X .

Definition 48.1.1 We define Γ− (S)≡ {ω ∈Ω : Γ(ω)∩S ̸= /0}

We will consider a theory of measurability of set valued functions. The followingtheorem is the main result in the subject. In this theorem the third condition is what we willrefer to as measurable.

Theorem 48.1.2 The following are equivalent in case of a complete σ finite measurespace. However 3 and 4 are equivalent for any measurable space consisting only of aset Ω and a σ algebra C .

1. For all B a Borel set in X ,Γ− (B) ∈ C .

2. For all F closed in X , Γ− (F) ∈ C

3. For all U open in X ,Γ− (U) ∈ C

4. There exists a sequence, {σn} of measurable functions satisfying σn (ω) ∈ Γ(ω)such that for all ω ∈Ω,

Γ(ω) = {σn (ω) : n ∈ N}

These functions are called measurable selections.

5. For all x ∈ X ,ω → dist(x,Γ(ω)) is a measurable real valued function.

6. G (Γ)≡ {(ω,x) : x ∈ Γ(ω)} ⊆ C ×B(X) .

Proof: It is obvious that 1.) ⇒ 2.). To see that 2.) ⇒ 3.) note that Γ− (∪∞i=1Fi) =

∪∞i=1Γ− (Fi) . Since any open set in X can be obtained as a countable union of closed sets,

this implies 2.) ⇒ 3.).Now we verify that 3.) ⇒ 4.). For convenience, drop the assumption that Γ(ω) is

closed in this part of the argument. It will just be set valued and satisfy the measurabilitycondition. A measurable selection will be obtained in Γ(ω). Let {xn}∞

n=1 be a countabledense subset of X . For ω ∈ Ω, let ψ1 (ω) = xn where n is the smallest integer such thatΓ(ω)∩B(xn,1) ̸= /0. Therefore, ψ1 (ω) has countably many values, xn1 ,xn2 , · · · where n1 <n2 < · · · . Now

{ω : ψ1 = xn}=

{ω : Γ(ω)∩B(xn,1) ̸= /0}∩ [Ω\∪k<n {ω : Γ(ω)∩B(xk,1) ̸= /0}] ∈ C .

Thus we see that ψ1 is measurable and dist(ψ1 (ω) ,Γ(ω))< 1. Let

Ωn ≡ {ω ∈Ω : ψ1 (ω) = xn} .

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