1590 CHAPTER 48. MULTIFUNCTIONS AND THEIR MEASURABILITY

Letting y = u, we use Fatou’s lemma to write

lim infn→∞

∫Ω

∫ T

0

(⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩+ c(t,ω)+C ||u(t,ω)||pV

)dtdP≥

∫Ω

∫ T

0lim inf

n→∞⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩+

(c(t,ω)+C ||u(t,ω)||pV

)dtdP

≥∫

Ω

∫ T

0

(c(t,ω)+C ||u(t,ω)||pV

)dtdP.

Here, we added the term c(t,ω)+C ||u(t,ω)||pV to make the integrand nonnegative in orderto apply Fatou’s lemma. Thus,

lim infn→∞

∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩dtdP≥ 0.

Consequently, using the claim in the last inequality,

0 ≥ lim supn→∞

⟨zn,un−u⟩V ′,V

≥ lim infn→∞

∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩dtdP

= lim infn→∞⟨zn,un−u⟩V ′,V

≥∫

Ω

∫ T

0lim inf

n→∞⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩dtdP≥ 0,

hence, we find thatlimn→∞⟨zn,un−u⟩V ′,V = 0. (48.7.41)

We need to show that if y is given in V then

lim infn→∞⟨zn,un− y⟩V ′,V ≥ ⟨z(y) ,u− y⟩ V ′,V , z(y) ∈ Âu

Suppose to the contrary that there exists y such that

η = lim infn→∞⟨zn,un− y⟩V ′,V < ⟨z,u− y⟩V ′,V , (48.7.42)

for all z ∈ Âu. Take a subsequence, denoted still with subscript n such that

η = limn→∞⟨zn,un− y⟩V ′,V

Note that this subsequence does not depend on (t,ω). Thus

limn→∞⟨zn,un− y⟩V ′,V < ⟨z,u− y⟩V ′,V (48.7.43)

We will obtain a contradiction to this. In what follows, we continue to use the subsequencejust described which satisfies the above inequality.