1566 CHAPTER 48. MULTIFUNCTIONS AND THEIR MEASURABILITY

Then since xε (ω) is contained in K (ω), a compact set, and the diameter of each sim-plex is less than 1, it follows that Aε

(xε

k (ω) ,ω)

is contained in

A(K (ω)+B(0,1),ω)

which is a compact set. Let Wε (ω) ∈ R2n+2n2be defined as follows.

Wε (ω) :=(

tε1 (ω) , · · · , tε

n (ω) ,xε0 (ω) , · · · ,xε

n (ω) ,xε (ω) ,Aε (xε

1 (ω) ,ω) · · ·Aεm (xεmn (ω) ,ω)

)Thus Wε has values in a compact subset of R2n+2n2

and is measurable. By Lemma 48.2.2there exists a subsequence ε (ω)→ 0 and a measurable function ω →W(ω) such that

Wε(ω) (ω)→W(ω) =

(t1 (ω) , · · · , tn (ω) ,x0 (ω) , · · · ,

xn (ω) ,x(ω) ,y1 (ω) , · · · ,yn (ω)

)as ε (ω)→ 0. Recall also that

Aε (xεk (ω) ,ω)⊆ A

(PK(ω)xε

k ,ω)

Now ∣∣PK(ω)xεk (ω)−xε (ω)

∣∣= ∣∣PK(ω)xεk (ω)−PK(ω)xε (ω)

∣∣≤ |xεk −xε |< ε

Both xε(ω)k (ω) and xε(ω) (ω) converge to x(ω) and so the above shows that also,

PK(ω)xεk (ω)→ x(ω)

Therefore,Aε(ω)

(xε(ω)

k (ω) ,ω)⊆ A(x(ω) ,ω)+B(0,r)

whenever ε (ω) is small enough. Since A(x(ω) ,ω) is closed, this implies

yk (ω) ∈ A(x(ω) ,ω) .

Since A(x(ω) ,ω) is convex,

n

∑k=1

tk (ω)yk (ω) ∈ A(x(ω) ,ω) .

Also, from the construction,

xε (ω) = Aε (xε (ω) ,ω)≡n

∑k=0

tεk (ω)Aε (xε

k (ω) ,ω)

so passing to the limit as ε (ω)→ 0, we get

x(ω) =n

∑k=0

tk (ω)yk (ω) ∈ A(x(ω) ,ω)

and this is the measurable fixed point.