68.3. A MULTIPLE INTEGRAL 2327

Now in the above construction, from now on, unless indicated otherwise, H = L2 (T )where the measure is ordinary Lebesgue measure on T = [0,T ] or [0,∞) or some otherinterval of time. However, it could be more general, but for the sake of simplicity let it beLebesgue measure. Generalities appear to be nothing but identifying that which works inthe case of Lebesgue measure. If µ≪m everything would work also. A careful descriptionof what kind of measures work is in [102]. Also, for A a Borel set having finite Lebesguemeasure,

W (A)≡W (XA) .

This is a random variable, and as explained earlier, since any finite set of these is nor-mally distributed, if all the sets are pairwise disjoint, the random variables are independentbecause the covariance is a diagonal matrix.

Definition 68.3.1 Let D ≡{(t1, · · · , tm) : ti = t j for some i ̸= j

}. This is called the diago-

nal set. Here D⊆ T m where T is an interval [0,T ). Assume T < ∞ here. Let

0 = τ0 < τ1 < · · ·< τk = T

Then this can be used to partition T m into sets of the form

[τ i1−1,τ i1)×·· ·× [τ im−1,τ im)

such that T m is the disjoint union of these. An off diagonal step function f is one which isof the form

f (t1, · · · , tm) =k

∑i1··· ,im

ai1,··· ,imX[τ i1−1,τ i1 )×···×[τ im−1,τ im ) (t1, · · · , tm)

where ai1,··· ,im = 0 if ip = iq. This would correspond to a diagonal term because it wouldresult in a repeated half open interval. Thus we assume all these are equal to 0. Thecollection of all such off diagonal step functions will be denoted as Em. The m correspondsto the dimension.

Definition 68.3.2 Let Im : Em→ L2 (Ω) be defined in the obvious way.

Im

(k

∑i1··· ,im

ai1,··· ,imX[τ i1−1,τ i1 )×···×[τ im−1,τ im ) (t1, · · · , tm)

)

≡k

∑i1··· ,im

ai1,··· ,im

m

∏p=1

(Bτ ip−Bτ ip−1

)Then Im is linear. If you had two different partitions, you could take the union of them bothand by letting coefficients be repeated on the smaller boxes, one can assume that a singlepartition is being used. This is why it is clear that Im is linear.

Definition 68.3.3 Let f ∈ L2 (T m) . The symetrization of f is given by

f̃ (t1, · · · , tm)≡1

m! ∑σ∈Sm

f (tσ1 , · · · , tσm)