64.6. WIENER PROCESSES, ANOTHER APPROACH 2213

The following lemma shows that the functions eW (h) are dense in Lp (Ω) for every p> 1.

Lemma 64.6.5 Let F be the σ algebra determined by the random variables W (h). IfX ∈ Lp (Ω,F ,P) , p > 1 and

∫Ω

XeW (h)dP = 0 for every h ∈ H, then X = 0.

Proof: Let h1, · · · ,hp be given. Then for ti ∈ R,

∑i

tihi ∈ H

and so since W is linear,∫Ω

Xet·W(h)dP = 0, W(h)≡ (W (h1) , · · · ,W (hp))

Now by Lemma 64.6.3,∫Ω

X+et·(W (h1),··· ,W(hp))dP =∫Rp

et·ydν+ (y)

where ν+ (B) = E (X+XB (W(h))) . From Lemma 64.6.4, this function of t is finite for allt ∈ Rp. Similarly, ∫

Ω

X−et·(W (h1),··· ,W(hp))dP =∫Rp

et·ydν− (y)

where ν− (B) = E (X−XB (W(h))). Thus for ν equal to the signed measure ν ≡ ν+−ν−,

f (t)≡∫Rp

et·ydν (y) = 0

for t ∈ Rp. Also ∫Ω

X+eit·(W(h))dP =∫Rp

eit·ydν+ (y)

with a similar formula holding for X−. Thus

f (t)≡∫Rp

et·ydν (y) ∈ C

is well defined for all t ∈ Cp. Consider∫Rp

et·ydν+ (y)

Is this function analytic in each tk? Take a difference quotient. It equals for h ∈ C,

∫Ω

X+

(e(t+hek)·(W(h))− et·(W(h))

)h

dP =∫

Ω

X+et·W(h)

(ehek·(W(h))−1

)h

dP

In case ek ·W(h) = 0 there is nothing to show. Assume then that this is not 0. Then thisequals ∫

Ω

X+ek·(W(h))et·W(h)

(ehek·(W(h))−1

)h(ek·(W(h)))

dP