Note1.pdf

Here is a rigorous proof of Fubini’s Theorem on the equality of double and iterated integrals. The present version is slightly more general than the one stated in the textbook. Fubini’s Theorem. Let f be an integrable function on the rectangle R D Œa; b  Œc; d . Rb Suppose that for each y 2 Œc; d , the integral a f .x; y/ dx exists and moreover Rb a f .x; y/ dx as a function of y is integrable on Œc; d . Then “ Z dZ b f dA D f .x; y/ dx dy: c

R

a

Of course a similar statement holds with the role of x; y changed. When f is continuous on the rectangle R, all the integrability assumptions hold automatically and we have “ Z dZ b Z bZ d f dA D f .x; y/ dx dy D f .x; y/ dy dx: c

R

a

a

c

The proof begins as follows. Take arbitrary partitions x0 D a < x1 <


< xm D b of Œa; b and y0 D c < y1 <


< yn D d of Œc; d . Let P be the partition of R into the mn sub-rectangles Rij D Œxi 1 ; xi   Œyj 1 ; yj . As usual, set xi D xi

xi

yj D yj

1

yj

1

mij D inf f

Mij D sup f:

Rij

Rij

Since mij  f .x; y/  Mij

for .x; y/ 2 Rij ;

the comparison property of the integral in dimension 1 shows that Z xi mij xi  f .x; y/ dx  Mij xi if y 2 Œyj xi

Summing over all i from 1 to m, we obtain Z b m m X X f .x; y/ dx  Mij xi mij xi  iD1

a

if y 2 Œyj

1 ; yj :

i D1

Applying comparison once more gives ! Z yj Z m X mij xi yj  i D1

1 ; yj :

1

yj

1

a

b

f .x; y/ dx dy 

m X

! Mij xi yj :

i D1

Summing up over j from 1 to n, we obtain Z dZ b n X m n X m X X mij xi yj  f .x; y/ dx dy  Mij xi yj : j D1 i D1

c

a

j D1 i D1

Thus, for every partition P of the rectangle R, Z dZ b L.f; P/  f .x; y/ dx dy  U.f; P/: c

a

On ’ the other hand, since by the assumption f is integrable on R, the double integral R f dA is the unique number which satisfies “ L.f; P/  f dA  U.f; P/ R

for every partition P. Therefore, we must have “ Z dZ b f dA D f .x; y/ dx dy; R

as claimed.

c

a