278 CHAPTER 9. MEASURES AND MEASURABLE FUNCTIONS

9.16 Exercises1. Using some form of Kuratowski’s theorem show the following: Let K (ω) be a closed

convex bounded subset of Rn where ω → K (ω) is a measurable multifunction. Letx→ f (x,ω) : K (ω)→ K (ω) be continuous for each ω and ω → f (x,ω) is mea-surable, meaning inverse images of sets open in Rn are in F where (Ω,F ) is ameasurable space. Then there exists x(ω) ∈ K (ω) such that ω → x(ω) is measur-able and f (x(ω) ,ω) = x(ω).

2. If you have K (ω) a closed convex nonempty set in Rn and also ω→ K (ω) is a mea-surable multifunction, show ω→ PK(ω)x is measurable where PK(ω) is the projectionmap which gives the closest point in K (ω). Consider Corollary 6.3.2 on Page 163 orTheorem 11.6.8 and Problem 10 on Page 152 to see the use of this projection map.Also you may want to use Theorem 9.15.2 involving the countable dense subset ofK (ω) consisting of measurable functions.

3. Let ω→K (ω) be a measurable multifunction inRp and let K (ω) be convex, closed,and compact for each ω . Let A(·,ω) : K (ω)→ Rp be continuous and ω→ A(x,ω)be measurable. Then if ω→y (ω) is measurable, there exists measurable ω→x(ω)such that for all z ∈ K (ω) ,

(y (ω)−A(x(ω) ,ω) ,z (ω)−x(ω))≤ 0

This is a measurable version of Browder’s lemma, a very important result in nonlin-ear analysis. Hint: You want to have for each ω ,

PK(ω) (y (ω)−A(x,ω)+x) = x

Use Problem 2 and the measurability of Brouwer fixed points discussed above.

4. In the situation of the above problem, suppose also that lim|x|→∞

(A(x,ω),x)|x| = ∞

Show that there exists measurable x(ω) such that A(x(ω) ,ω) = y (ω). Hint: Letxn (ω) be the solution of Problem 3 in which Kn = B(0,n). Show that these arebounded for each ω . Then use Corollary 9.15.6 to get x(ω) , a suitable limit suchthat A(x(ω) ,ω) = y (ω).