Page 88 - Vector Analysis

84 CHAPTER 3. Multiple Integrals

ż

f (x, ¨) : B Ñ R is denoted by f (x, y) dy, and the upper integral of f (x, ¨) : B Ñ R is

B

ż

denoted by f (x, y) dy. If for each x P A the upper integral and the lower integral of

Bż

f (x, ¨) : B Ñ R are the same, we simply write f (x, y) dy for the integrals of f (x, ¨) over

B

żż ż

B. The integrals f (x, y) dx, f (x, y) dx and f (x, y) dx are defined in a similar way.

AA A

Theorem 3.25 (Fubini’s Theorem). Let A Ď Rn and B Ď Rm be bounded sets, and f :

A ˆ B Ñ R be bounded. For x P Rn and y P Rm, write z = (x, y). Then

ż ż (ż ) ż (ż )ż

f (z) dz ď f (x, y)dy dx ď f (x, y)dy dx ď f (z) dz (3.3)

AˆB AB AB AˆB

and

ż ż (ż ) ż (ż )ż

f (z) dz ď f (x, y)dx dy ď f (x, y)dx dy ď f (z) dz . (3.4)

AˆB BA BA AˆB

In particular, if f : A ˆ B Ñ R is Riemann integrable, then

ż ż (ż ) ż (ż )

f (z) dz = f (x, y)dy dx = f (x, y)dy dx

AˆB AB AB

ż (ż ) ż (ż )

= f (x, y)dx dy = f (x, y)dx dy .

BA BA

Proof. It suffices to prove (3.3). Let ε ą 0 be given. Choose a partition P of A ˆ B such

ż

that L(f, P) ą f (z) dz ´ ε. Since P is a partition of A ˆ B, there exist partition Px

AˆB

of A and partition Py of B such that P = ␣∆ = R ˆ S ˇ R P Px, S P Py (. By Proposition
ˇ

3.14 and Corollary 3.16, we find that

ż (ż )ż (ż )

f (x, y) dy dx = 1A(x) f (x, y)1B(y) dy dx

AB ż (Ť R Ť PPy S
RPPx
S )

ż

ÿ ÿ f AˆB(x, y) dy dx
ě

R PPx R S PPy S
ż (ż )
ÿÿ f AˆB(x, y) dy dx
ě

R PPx S PPy R S

ÿ inf f AˆB(x, y)νm(S)νn(R)
ě
R PPx,S PPy (x,y)PRˆS

ż

ÿ inf f AˆB(x, y)νn+m(∆) = L(f, P) ą f (z)dz ´ ε .
=

∆PP (x,y)P∆ AˆB