48.3. A SET VALUED BROWDER LEMMA WITH MEASURABILITY 1559

each cnk being in V . We can assume that ∥un (ω)∥ ≤ 2∥u(ω)∥ for all ω . Then, by as-

sumption, there is a measurable selection for ω→ A(cn

k ,ω)

denoted as ω→ ynk (ω) . Thus,

ω → ynk (ω) is measurable into V ′ and yn

k (ω) ∈ A(cn

k ,ω)

for all ω ∈Ω. Consider now,

yn (ω) =mn

∑k=1

ynk (ω)XEn

k(ω) .

It is measurable and for ω ∈ Enk it equals yn

k (ω) ∈ A(cn

k ,ω)= A(un (ω) ,ω) . Thus, yn

is a measurable selection of ω → A(un (ω) ,ω) . By the assumption A(·,ω) is bounded,for each ω these yn (ω) lie in a bounded subset of V ′. The bound might depend on ω

of course. It follows now from Lemma 48.2.2 that there is a subsequence{

yn(ω)}

thatconverges weakly to y(ω), where ω → y(ω) is measurable. But,

yn(ω) (ω) ∈ A(un(ω) (ω) ,ω

)is a convex closed set for which u→ A(u,ω) is upper-semicontinuous and un(ω) → u,hence, y(ω) ∈ A(u(ω) ,ω). This is the claimed measurable selection.

The next lemma is about the projection map onto a set valued map whose values areclosed convex sets.

Lemma 48.3.2 Let ω →K (ω) be measurable into Rn where K (ω) is closed and con-vex. Then ω → PK (ω)u(ω) is also measurable into Rn if ω → u(ω) is measurable. HerePK (ω) is the projection map giving the closest point.

Proof: It follows from standard results on measurable multi-functions [70] also inTheorem 48.1.2 above that there is a countable collection {wn (ω)} , ω → wn (ω) beingmeasurable and wn (ω) ∈K (ω) for each ω such that for each ω, K (ω) = ∪nwn (ω). Let

dn (ω)≡min{∥u(ω)−wk (ω)∥ ,k ≤ n}

Let u1 (ω)≡ w1 (ω) . Letu2 (ω) = w1 (ω)

on the set{ω : ∥u(ω)−w1 (ω)∥< {∥u(ω)−w2 (ω)∥}}

andu2 (ω)≡ w2 (ω) off the above set.

Thus ∥u2 (ω)−u(ω)∥= d2. Let

u3 (ω) = w1 (ω) on{

ω : ∥u(ω)−w1 (ω)∥<∥∥u(ω)−w j (ω)

∥∥ , j = 2,3

}≡ S1

u3 (ω) = w2 (ω) on S1∩{

ω : ∥u(ω)−w1 (ω)∥<∥∥u(ω)−w j (ω)

∥∥ , j = 3

}u3 (ω) = w3 (ω) on the remainder of Ω