65.8. LOCALIZATION FOR ELEMENTARY FUNCTIONS 2251

Is this in B ([a, t0])×Ft0? This is what needs to be checked. Since A is progressivelymeasurable,

A−1 (O)∩ [a, t0]×V ×Ω ∈B ([a, t0])×B (V )×Ft0 ≡Pt0

because A−1 (O) is a progressively measurable set. So let

G ≡{

S ∈Pt0 : {(t,ω) ∈ [a, t0]×Ω : X (t,ω) ∈ Stω} ∈B ([a, t0])×Ft0

}It is clear that G contains the π system composed of sets of the form I×B×W where I isan interval in [a, t0], B is Borel, and W ∈Ft0 . This is because for S of this form, Stω = B or/0. Thus if not empty,

{(t,ω) ∈ [a, t0]×Ω : X (t,ω) ∈ Stω}= X−1 (B)∩ [0, t0]×Ω ∈B ([a, t0])×Ft0

because X is given to be progressively measurable. Now if S∈ G , what about SC? You have(SC)

tω = (Stω)C thus {

(t,ω) ∈ [a, t0]×Ω : X (t,ω) ∈(SC)

tω

}=

{(t,ω) ∈ [a, t0]×Ω : X (t,ω) ∈ (Stω)

C}

which is the complement with respect to [a, t0]×Ω of a set in B ([a, t0])×Ft0 . Therefore,G is closed with respect to complements. It is clearly closed with respect to countabledisjoint unions. It follows, G = Pt0 . Thus

{(t,ω) ∈ [a, t0]×Ω : X (t,ω) ∈ Stω} ∈B ([a, t0])×Ft0

where S = A−1 (O)∩ [a, t0]×V ×Ω. In other words,

{(t,ω) , t ≤ t0 : A(t,X (t,ω) ,ω) ∈ O} ∈B ([0, t0])×Ft0

and so (t,ω)→ A(t,X (t,ω) ,ω) is progressively measurable.

65.8 Localization For Elementary FunctionsIt is desirable to extend everything to stochastically square integrable functions. This willinvolve localization using a suitable stopping time. First it is necessary to understand lo-calization for elementary functions. As above, we are in the situation described by thefollowing diagram.

U↓ Q1/2

U1 ⊇ JQ1/2U J←1−1

Q1/2U

Φn ↘ ↓ Φ

H