68.3. A MULTIPLE INTEGRAL 2333
closed with respect to scalar multiplication. Hence it is a subspace of the vector space ofall functions and it is therefore, a vector space.
Following [102], for f one of these elementary functions,
f (t1, · · · , tn)≡∑i
ciXAi1×···×Ain(t1, · · · , tn)
where if any two indices are repeated, then ci = 0, and the Ak are all disjoint,
In ( f )≡∑i
ciW (Ai1) · · ·W (Ain)
Lemma 68.3.9 In is linear on En. If f ∈ En and σ is a permutation of (1, · · · ,n) and
fσ (t1, · · · , tn)≡ f(tσ(1), · · · , tσ(tn)
),
and f is symmetric, thenIn ( fσ ) = In ( f )
For f = ∑i ciXAi1×···×Ain, one can conclude that
In ( f ) = n! ∑i1<i2<···<in
ci1,··· ,in ∏W (Ain) (68.3.15)
Also, the following holds for the expectation. For f ,g ∈ En,Em respectively,
E (In ( f ) Im (g)) ={
0 if n ̸= mn!(
f̃ , g̃)
L2(T n)if n = m
where f̃ denotes the symetrization of f given by
f̃ (t1, · · · , tn)≡1n! ∑
σ∈Sn
f(tσ(1), · · · , tσ(n)
)Proof: It is clear from the definition being well defined that In is linear. In particular,
consider
In
(a∑
iciXAi1×···×Ain
+b∑j
djXB j1×···×B jn
).
As explained above in the observation that En is a vector space, it can be assumed that theAik and B jk are all from a single set of disjoint Borel sets of T . Then the above is of theform
a∑i
ci ∏k
W(Aik
)+b∑
jdj ∏
kW(B jk
)= aIn
(∑
iciXAi1×···×Ain
)+bIn
(∑
jdjXB j1×···×B jn
)