48.2. EXISTENCE OF MEASURABLE FIXED POINTS 1553

Thus, from now on, simply denote as n the upper limit and let ω ∈ Ωn. If fr (xr,ω) = xrand

xr =n

∑i=1

aiyi (ω)

for ∑ni=1 ai = 1 and the yi described in the above lemma, we need

fr (xr,ω) ≡n

∑i=1

yi (ω)ψ i ( f (xr,ω) ,ω)

=n

∑j=1

y j (ω)ψ j

(f

(n

∑i=1

aiyi,ω

),ω

)=

n

∑j=1

a jy j (ω) = xr.

Also, if this is satisfied, then we have the desired fixed point.This will be satisfied if for each j = 1, · · · ,n,

a j = ψ j

(f

(n

∑i=1

aiyi,ω

),ω

)(48.2.3)

so, let

Σn−1 ≡

{a ∈ Rn :

n

∑i=1

ai = 1, ai ≥ 0

}and let h(·,ω) : Σn−1→ Σn−1 be given by

h(a,ω) j ≡ ψ j

(f

(n

∑i=1

aiyi,ω

),ω

)

Can we obtain a fixed point a(ω) such that ω → a(ω) is measurable? Since h(·,ω) is acontinuous function of a and ω → h(x,ω) is measurable, such a measurable fixed pointexists thanks to Theorem 48.2.4 or the much easier Theorem 48.2.8 above. Then xr (ω) =

∑ni=1 ai (ω)yi (ω) so xr is measurable.

The following is the Schauder fixed point theorem for measurable fixed points.

Theorem 48.2.13 Let ω → K (ω) be a measurable multifunction which has convex andclosed values in a separable Banach space. Let f (·,ω) : K (ω)→ K (ω) be continuousand ω → f (x,ω) is measurable and f (K (ω) ,ω) is compact. Then f (·,ω) has a fixedpoint x(ω) such that ω → x(ω) is measurable.

Proof: Recall that f (xr (ω) ,ω)− fr (xr (ω) ,ω) ∈ B(0,r) and fr (xr (ω) ,ω) = xr (ω)with xr (ω) ∈ convex hull of f (K (ω) ,ω) ⊆ K (ω) . Here xr is measurable. By Lemma48.2.2 there is a measurable function x(ω) which equals the weak limr(ω)→0 xr(ω) (ω) .

However, since f (K (ω) ,ω) is compact, there is a subsequence still denoted with r (ω)such that f

(xr(ω),ω

)converges strongly to some x ∈ f (K (ω) ,ω). It follows that

fr(ω)

(xr(ω) (ω) ,ω

)