67.9. SOME REPRESENTATION THEOREMS 2311

RecallGt ≡ σ (W(u)−W(r) : 0≤ r < u≤ t)

It suffices to consider only t and u,r in a countable dense subset of R denoted as D. Thisfollows from continuity of the Wiener process. To see this, let 0≤ r < u≤ t U be open andUn ↑U where each Un is open and Un ⊆Un+1,∪nUn =U . Then letting un ↑ u and rn ↑ r,unrnbeing in the countable dense set,

(W(u)−W(r))−1 (Un) ⊆ ∪∞k=1∩ j≥k (W(u j)−W(r j))

−1 (Un)

⊆ (W(u)−W(r))−1 (Un)

and so

(W(u)−W(r))−1 (U) = ∪n (W(u)−W(r))−1 (Un)

⊆ ∪∞n=1∪∞

k=1∩ j≥k (W(u j)−W(r j))−1 (Un)

⊆ ∪n (W(u)−W(r))−1 (Un)= (W(u)−W(r))−1 (U)

Now the set in the middle which has two countable unions and a countable intersection isin

σ (W(u)−W(r) : 0≤ r < u≤ t,r,u ∈ D)

Thus in particular, one would get the same filtration from

Gt = σ (W(u)−W(r) : 0≤ r < u≤ t,r,u ∈ D)

Since W(0) = 0, this is the same as

Gt = σ (W(u) : 0≤ u≤ t,u ∈ D)

Lemma 67.9.8 Random variables of the form

φ (W(t1) , · · · ,W(tk)) , φ ∈C∞c

(Rk)

are dense in L2 (Ω,GT ,P) where t1 < t2 · · ·< tk is a finite increasing sequence of

(Q∪{T})∩ [0,T ] .

Proof: Let g ∈ L2 (Ω,GT ,P) . Also let{

t j}∞

j=1 be the points of (Q∪{T})∩ [0,T ] . Let

Gm ≡ σ (W(tk) : k ≤ m)

Thus the Gm are increasing but each is generated by finitely many W(tk). Also as explainedabove,

GT = σ (W(u) : 0≤ u≤ T,u ∈ (Q∪{T})∩ [0,T ])= σ (Gm,m < ∞) .