Kenneth Kuttler

450 CHAPTER 16. APPROXIMATION OF FUNCTIONS AND THE INTEGRAL

Consider the iterated integral∫ b1

a1· · ·∫ bp

apαxα1

1 · · ·xα pp dxp · · ·dx1. It means just what it

meant in calculus. You do the integral with respect to xp first, keeping the other variablesconstant, obtaining a polynomial function of the other variables. Then you do this one withrespect to xp−1 and so forth. Thus, doing the computation, it reduces to

∏k=1

(∫ bk

xαkk dxk

)= α

∏k=1

(bαk+1

αk +1− aαk+1

αk +1

)and the same thing would be obtained for any other order of the iterated integrals. Sinceeach of these integrals is linear, it follows that if (i1, · · · , ip) is any permutation of (1, · · · , p) ,then for any polynomial q,∫ b1

· · ·∫ bp

q(x1, ...,xp)dxp · · ·dx1 =∫ bi1

aip

· · ·∫ bip

aip

q(x1, ...,xp)dxip · · ·dxi1

Now let f : ∏pk=1 [ak,bk]→ R be continuous. Then each iterated integral results in a con-

tinuous function of the remaining variables and so the iterated integral makes sense. Forexample, by Proposition 16.4.5,

∣∣∣∫ dc f (x,y)dy−

∫ dc f (x̂,y)dy

∣∣∣=∣∣∣∣∫ d

c( f (x,y)− f (x̂,y))dy

∣∣∣∣≤ maxy∈[c,d]

| f (x,y)− f (x̂,y)|< ε

if |x− x̂| is sufficiently small, thanks to uniform continuity of f on the compact set [a,b]×[c,d]. Thus it makes perfect sense to consider the iterated integral

∫ ba∫ d

c f (x,y)dydx. Thenusing Proposition 16.4.5 on the iterated integrals along with Theorem 16.2.1, there exists asequence of polynomials which converges to f uniformly {pn} . Then applying Proposition16.4.5 repeatedly,∣∣∣∣∣

∫ bi1

aip

· · ·∫ bip

aip

f (x)dxp · · ·dx1−∫ bi1

aip

· · ·∫ bip

aip

pn (x)dxp · · ·dx1

∣∣∣∣∣≤ ∥ f − pn∥

∏k=1|bk−ak| (16.4)

With this, it is easy to prove a rudimentary Fubini theorem valid for continuous functions.

Theorem 16.4.6 f : ∏pk=1 [ak,bk]→ R be continuous. Then for (i1, · · · , ip) any permuta-

tion of (1, · · · , p) ,∫ bi1

aip

· · ·∫ bip

aip

f (x)dxip · · ·dxi1 =∫ b1

· · ·∫ bp

f (x)dxp · · ·dx1

If f ≥ 0, then the iterated integrals are nonnegative if each ak ≤ bk.

Proof: Let ∥pn− f∥→ 0 where pn is a polynomial. Then from 16.4,∫ bi1

ai1

· · ·∫ bip

aip

f (x)dxip · · ·dxi1 = limn→∞

∫ bi1

aip

· · ·∫ bip

aip

pn (x)dxip · · ·dxi1

450 CHAPTER 16. APPROXIMATION OF FUNCTIONS AND THE INTEGRALConsider the iterated integral I Vee f a ax! ...x5? dxpy---dx1. It means just what itmeant in calculus. You do the integral with respect to x, first, keeping the other variablesconstant, obtaining a polynomial function of the other variables. Then you do this one withrespect to x»; and so forth. Thus, doing the computation, it reduces toP by a P perl qh!a kd =a —II(/, Xk ve) tes —)and the same thing would be obtained for any other order of the iterated integrals. Sinceeach of these integrals is linear, it follows that if (i),--- ,i,) is any permutation of (1,---,p),then for any polynomial q,by bp iy ip[- / (x1,-. 3X p)AdXp-- -dx, = [ of q(xX1,-- 1X p) AX, ** dx},Now let f : []?_, [ax,x] + R be continuous. Then each iterated integral results in a con-tinuous function of the remaining variables and so the iterated integral makes sense. For(x,y)dy— JF &y)dy| =example, by Proposition 16.4.5,[vere dy < max f(y) —F y)] <eye [c,d]if |x —%| is sufficiently small, thanks to uniform continuity of f on the compact set [a,b] x[c,d]. Thus it makes perfect sense to consider the iterated integral p fi? f (x,y) dydx. Thenusing Proposition 16.4.5 on the iterated integrals along with Theorem 16.2.1, there exists asequence of polynomials which converges to f uniformly {p,}. Then applying Proposition16.4.5 repeatedly,bi dip Di, Dip/ eee / f (a) dXxp eae dx — / eee Pn (a) dXxp eae dxa, a. a.’ ip ’ ip So ip “dipPS lf — Pall [] \bx — el (16.4)k=lWith this, it is easy to prove a rudimentary Fubini theorem valid for continuous functions.Theorem 16.4.6 f : []/_, [ax,b«] > R be continuous. Then for (i,,+++ ,ip) any permuta-tion of (1,++- sP);iy " F by ”[ ane x) dx;, °° -dxi, = [- ofr x) AdXp- -- dx,dipIf f = 0, then the iterated integrals are nonnegative if each ax < Db.Proof: Let ||p, — f|| — 0 where p, is a polynomial. Then from 16.4,iy i Dip[~ (x) dx;, -+-dxi, = Jim of Pn (@) dx;, Xi,ipip