30.3. FILTRATIONS 821

Lemma 30.3.7 Define Q+h as

Q+h≡ {(t +h,ω) : (t,ω) ∈ Q} .

Then if Q ∈P, it follows that Q+h ∈P .

Proof: This is most easily seen through the use of the following diagram. In this dia-gram, Q is in P so it is progressively measurable.

QS Q+h

tt−hBy definition, S in the picture is B ((−∞, t−h])×Ft−h measurable. Hence S+ h ≡

Q+ h∩Ω× (−∞, t] is B ((−∞, t])×Ft−h measurable. To see this, note that if B×A ∈B ((−∞, t−h])×Ft−h, then translating it by h gives a set in B ((−∞, t])×Ft−h. Then ifG consists of sets S in B ((−∞, t−h])×Ft−h for which S+ h is in B ((−∞, t])×Ft−h,G is closed with respect to countable disjoint unions and complements. Thus, G equalsB ((−∞, t−h])×Ft−h. In particular, it contains the set S just described. ■

Now for h > 0,

τh f (t)≡{

f (t−h) if t ≥ h,0 if t < h. .

Lemma 30.3.8 Let Q ∈P. Then τhXQ is P measurable.

Proof: If τhXQ (t,ω) = 1, then you need to have (t−h,ω) ∈ Q and so (t,ω) ∈ Q+h.Thus

τhXQ = XQ+h,

which is P measurable since Q ∈P . In general,

τhXQ = X[h,T ]×ΩXQ+h,

which is P measurable. ■This lemma implies the following.

Lemma 30.3.9 Let f (t,ω) have values in a separable Banach space and suppose f isP measurable. Then τh f is P measurable.

Proof: Taking values in a separable Banach space and being P measurable, f is thepointwise limit of P measurable simple functions. If sn is one of these, then from the abovelemmas, τhsn is P measurable. Then, letting n→ ∞, it follows that τh f is P measurable.■

The following is similar to Proposition 30.1.2. It shows that under pretty weak condi-tions, an adapted process has a progressively measurable adapted version.

Proposition 30.3.10 Let X be a stochastically continuous adapted process for a nor-mal filtration defined on a closed interval, I ≡ [0,T ]. Then X has a progressively measur-able adapted version.