186 CHAPTER 8. POSITIVE LINEAR FUNCTIONALS

∫ R

−R

∣∣ f (x1, · · · ,xp−1,xp)− f (x1, · · · , x̂p−1,xp)∣∣dxp <

ε

2R2R = ε

and so xp−1→∫

∞

−∞f (x1, · · · ,xp−1,xp)dxp is continuous and zero off some interval and so

it is integrable. Continuing this way shows that the functional defined above makes perfectsense. You can keep doing the iterated integrals. ■

The idea of the following Lemma is in Problem 6. You could use the result of thatproblem for transpositions obtain the conclusion of the following lemma by consideringthe product of transpositions.

Lemma 8.0.3 If L f is given in 8.1 and if (i1, · · · , ip) is any permutation of (1, · · · , p)with σ being the name of this permutation, then defining

Lσ f ≡∫

∞

−∞

· · ·∫

∞

−∞

f (x1, · · · ,xp)dxipdxip−1 · · ·dxi1

it follows that Lσ = Lα on Cc (Rp) where α is any other permutation.

Proof: Let Tn denote a tiling of Rp into disjoint half open rectangles, each of diameter1/2n. Let ∏

pi=1[0,2

−n) be in Tn to be specific, thus forcing each Q in Tn to be the unionof the Q in Tn+1. Also denote by Qr = ∏

pi=1[ri,ri + 2−n) one of the half open rectangles

so described and letting V be the set of such vertices r ≡ (r1, · · · ,rp). Then using themean value theorem for one dimensional integrals in the successive iterated integrals, (SeeProblem 5 on Page 132) it follows that

Lσ f ≡ ∑r∈V

∫ ri1+2−n

ri1

· · ·∫ rip+2−n

rip

f (x1, · · · ,xp)dxipdxip−1 · · ·dxi1 = ∑r∈V

(2−n)p f (xrσ )

there being only finitely many terms in the above sum and xrσ is a point of Qr. Since fhas compact support, there is a positive integer m such that the support of f is containedin ∏

pi=1[−m,m). Thus the only r in the above sum are those for which ri = k2−n for k an

integer in [−m2n,m2n).By uniform continuity of f there is δ such that if |x−y| < δ , then | f (x)− f (y)| <

ε/(2mp)p. Then by choosing n large enough so that each Qr has diameter less than δ , iffollows that

|Lσ f −Lα f |<m2n

∑k1=−m2n

· · ·m2n

∑kp=−m2n

(2−n)p

ε = (2mp2n)p (2−n)p ε

(2mp)p = ε

Therefore, since ε is arbitrary, Lσ = Lα for any two permutations σ ,α . ■

8.1 Partitions of UnityThe support of a function f , denoted as spt( f ), is the closure of the set on which thefunction is nonzeo.

Definition 8.1.1 Define Cc (X) to be the functions which have complex values andcompact support. This means spt( f ) ≡ {x ∈ X : f (x) ̸= 0} is a compact set. Then L :Cc (X)→ C is called a positive linear functional if it is linear and if, whenever f ≥ 0,then L( f ) ≥ 0 also. When f is a continuous function and spt( f ) ⊆ V an open set, we sayf ∈Cc (V ). Here X is some metric space.