2214 CHAPTER 64. WIENER PROCESSES

Now ∣∣∣∣ez−1z

∣∣∣∣=∣∣∣∣∣1z ∞

∑k=1

zk

k!

∣∣∣∣∣≤ ∞

∑k=0≤ |ez|

and so the integrand is dominated by∣∣∣∣∣∣X+ek·(W(h))et·W(h)

(ehek·(W(h))−1

)h(ek·(W(h)))

∣∣∣∣∣∣ ≤ X+∣∣∣ek·(W(h))et·W(h)eh(ek·(W(h)))

∣∣∣= X+

∣∣∣ek·(W(h))e(t+hek)·W(h)∣∣∣

From Lemma 64.6.4 which says that eW (h) is in Lq (Ω) for each q > 1, this is in particulartrue for q = mp where m is an arbitrary positive integer satisfying

p >m+1

m

Then the integrand is of the form f gh where f ∈ Lp and gh is bounded in Lmp. Therefore,

α ≡ (pm)/(m+1)> 1

and ∫Ω

| f gh|α dP =∫

Ω

| f |α |gh|α dP≤(∫

Ω

| f |p dP)m/(m+1)(∫

Ω

|gh|pm dP)1/(m+1)

which is bounded. By the Vitali convergence theorem,

limh→0

∫Ω

X+

(e(t+hek)·(W(h))− et·(W(h))

)h

dP =∫

Ω

X+ek·(W(h))et·W(h)dP

and so this function of tk is analytic. Similarly one can do the same thing for the integralinvolving X−. Thus

0 =∫Rp

et·ydν (y)

whenever t j ∈R for all j and t1→∫Rp et·ydν (y) is analytic onC. Thus this analytic function

of t1 is zero for all t1 ∈ C since it is zero on a set which has a limit point, and in particular∫Rp

eit1y1+t2y2+···+tpypdν (y) = 0

where each t j is real. Now repeat the argument with respect to t2 and conclude that∫Rp

eit1y1+it2y2+···+tpypdν (y) = 0,

and continue this way to conclude that

0 =∫Rp

eit·ydν (y)