76.3. THE EXISTENCE OF APPROXIMATE SOLUTIONS 2543

φ ∈ V ′ and the positive and negative parts of this function. Each of these is the limit ofa sequence of simple functions involving combinations of indicator functions of the formXQ. Thus τhφ ( f ) = φ (τh f ) is the limit of simple functions involving combinations offunctions τhXQ and, as just shown, these simple functions are progressively measurable.Thus τh f is also progressively measurable by the Pettis theorem.

This Lemma states that you can do τh to progressively measurable functions and endup with one which is progressively measurable. Let

B ∈L(W,W ′

)satisfy

⟨Bx,y⟩= ⟨By,x⟩ , ⟨Bx,x⟩ ≥ 0 (76.3.7)

Also suppose that

A is monotone and hemicontinuous from V to V ′ (76.3.8)

This means the operator is monotone:

⟨Au−Au,u− v⟩V ′,V ≥ 0

and hemicontinuous:limt→0⟨A(u+ tv) ,w⟩V ′,V = ⟨Au,w⟩V ′,V

Also we assume that A is bounded and takes the form

Au(t,ω) = A(t,u(t,ω) ,ω)

for u ∈ V . Such an operator is type M and this is what we use. Such an operator is definedby:

If un→ u weakly in V and Aun→ ξ weakly in V ′ and lim supn→∞

⟨Aun,un⟩ ≤ ⟨ξ ,u⟩

Then the above impliesAu = ξ .

We define Vω as Lp (0,T,V ) with the definition of V ′ω similar, the subscript denotingthat ω is fixed, the σ algebra of measurable sets being the Borel sets, B ([0,T ]). Also,

(t,u,ω)→ A(t,u,ω) (76.3.9)

is progressively measurable.Suppose A(ω) is monotone and hemicontinuous and bounded from Vω to V ′ω . Thus

A(ω) is type M from Vω to V ′ω (76.3.10)

whereA(ω)u≡ A(t,u,ω)

We assume the estimates found in the next lemma.