9.15. MULTIFUNCTIONS AND THEIR MEASURABILITY 275

measure space. Suppose ψω : E (ω)×Ω → R has the property that x → ψω (x,ω) iscontinuous and ω → ψω (x(ω) ,ω) is measurable if x(ω) ∈ E (ω) and ω → x(ω) is mea-surable (x(ω) a measurable selection of E (ω)). Then there exists a measurable func-tion f with f (ω) ∈ E (ω) such that ψω ( f (ω) ,ω) = maxx∈E(ω) ψω (x,ω) . Furthermore,ω → ψω ( f (ω) ,ω) is measurable.

Proof: Let C (ω) = {ei (ω)}∞

i=1 be a countable dense subset of E (ω) with each ei (ω)measurable. This countable dense subset exists by Theorem 9.15.2. Let

Cn (ω)≡ {e1 (ω) , ...,en (ω)} .

Let ω → fn (ω) be measurable and satisfy

ψω ( fn (ω) ,ω) = supx∈Cn(ω)

ψ (x,ω) .

This is easily done as follows. Let

Bk ≡{

ω : ψω (ek (ω) ,ω)≥ ψω (e j (ω) ,ω) for all j ̸= k}.

Then let A1 ≡ B1 and if A1, ...,Ak have been chosen, let Ak+1 ≡ Bk+1 \(∪k

j=1Bk

). Thus

each Ak is measurable, and you let fn (ω) ≡ ek (ω) for ω ∈ Ak, so fn (ω) ∈ E (ω) andfn is measurable. Using Corollary 9.15.6, there is measurable f (ω) and a subsequencen(ω)≥ n such that fn(ω) (ω)→ f (ω) . Then by continuity,

ψω ( f (ω) ,ω) = limn(ω)→∞

ψω

(fn(ω) (ω) ,ω

)and this is an increasing sequence in this limit. Hence

ψω ( f (ω) ,ω)≥ supx∈Cn(ω)

ψω (x,ω)

for each n and so

ψω ( f (ω) ,ω)≥ supx∈C(ω)

ψω (x,ω) = supx∈E(ω)

ψω (x,ω) .

Since f is measurable, it follows by assumption, that ω→ψω ( f (ω) ,ω) is measurable. ■Note the following: If you have the simpler situation where ψ (x,ω) defined on X×Ω

with x→ ψ (x,ω) continuous and ω → ψ (x,ω) measurable but E (ω) a compact mea-surable multifunction as above, then the conditions will hold because you would haveω→ψ (x(ω) ,ω) is measurable if x(ω) is. Indeed, x(ω) is the limit of a sequence {xn (ω)}such that xn has finitely many values on measurable sets, Theorem 9.1.7. Hence, by conti-nuity, ψ (x(ω) ,ω) = limn→∞ ψ (xn (ω) ,ω) and since ω→ψ (xn (ω) ,ω) is measurable, sois ψ (x(ω) ,ω).

9.15.4 Measurability of Fixed PointsAs an interesting application is a consideration of the existence of measurable Brouwerfixed points. This is really quite amazing since Brouwer fixed points are not obtained as thelimit of a sequence of iterates although the above Sperner’s lemma algorithm provides an