48.2. EXISTENCE OF MEASURABLE FIXED POINTS 1551

Corollary 48.2.10 Let K (ω) be a closed convex bounded subset of Rn and let ω→ K (ω)be a measurable multifunction for ω ∈ Ω with (Ω,F ) a measurable space. Let f(·,ω) :K (ω)→ K (ω) and x→ f(x,ω) continuous on Rn. Suppose also ω → f(x,ω) is mea-surable for each fixed x. Then there exists a measurable fixed point x(ω) , f(x(ω) ,ω) =x(ω) ,x(ω) ∈ K (ω) .

48.2.4 Measurability Of Schauder Fixed PointsNow we consider the Schauder fixed point theorem. Let ω → K (ω) be a measurablemultifunction having closed convex values. Here K (ω) ⊆ X a separable Banach space.Also assume

f (·,ω) is continuous, f (·,ω) : K (ω)→ K (ω) ,

ω → f (x,ω) is measurable

Next we have the following approximation result.

Lemma 48.2.11 Let f (·,ω) be as above and f (K (ω) ,ω) ⊆ K (ω) for K (ω) convex andclosed and ω → K (ω) a measurable mutifunction. Suppose also that f (K (ω) ,ω) is acompact set. For each r > 0 and ω, there exists a finite set of points{

y1 (ω) , · · · ,yn(ω) (ω)}⊆ f (K (ω) ,ω),ω → yi (ω) measurable

and continuous functions ψ i (·,ω) defined on f (K (ω) ,ω) such that for y ∈ f (K (ω) ,ω),

n(ω)

∑i=1

ψ i (y,ω) = 1, (48.2.1)

ψ i (y,ω) = 0 if y /∈ B(yi (ω) ,r) , ψ i (y,ω)> 0 if y ∈ B(yi (ω) ,r) .

If

fr (x,ω)≡n(ω)

∑i=1

yi (ω)ψ i ( f (x,ω) ,ω), (48.2.2)

then whenever x ∈ K (ω),∥ f (x,ω)− fr (x,ω)∥ ≤ r.

Proof: Using the compactness of f (K (ω) ,ω), Proposition 48.1.6 says there exist mea-surable functions yi (ω){

y1 (ω) , · · · ,yn(ω) (ω)}⊆ f (K (ω) ,ω)⊆ K (ω)

such that{B(yi (ω) ,r)}n

i=1

covers f (K,ω). Letφ i (y,ω)≡ (r−∥y− yi (ω)∥)+