324 CHAPTER 12. THE CONSTRUCTION OF MEASURES

This definition is well defined because of Theorem 12.10.11.

Theorem 12.10.13 If A ∈S , B ∈F, then (µ ×λ )(A×B) = µ(A)λ (B), and µ ×λ is ameasure on S ×F called product measure.

Proof: The first assertion about the measure of a measurable rectangle was establishedabove. Now suppose {Ei}∞

i=1 is a disjoint collection of sets of S ×F . Then using themonotone convergence theorem along with the observation that (Ei)x∩ (E j)x = /0,

(µ×λ )(∪∞i=1Ei) =

∫X

λ ((∪∞i=1Ei)x)dµ

=∫

Xλ (∪∞

i=1 (Ei)x)dµ =∫

X

∞

∑i=1

λ ((Ei)x)dµ

=∞

∑i=1

∫X

λ ((Ei)x)dµ

=∞

∑i=1

(µ×λ )(Ei)

This proves the theorem.The next theorem is one of several theorems due to Fubini and Tonelli. These theorems

all have to do with interchanging the order of integration in a multiple integral.

Theorem 12.10.14 Let f : X×Y → [0,∞] be measurable with respect to S ×F and sup-pose µ and λ are σ finite. Then∫

X×Yf d(µ×λ ) =

∫X

∫Y

f (x,y)dλdµ =∫

Y

∫X

f (x,y)dµdλ (12.10.50)

and all integrals make sense.

Proof: For E ∈S ×F,∫Y

XE(x,y)dλ = λ (Ex),∫

XXE(x,y)dµ = µ(Ey).

Thus from Definition 12.10.12, 12.10.50 holds if f = XE . It follows that 12.10.50 holdsfor every nonnegative simple function. By Theorem 11.3.9 on Page 241, there exists anincreasing sequence, { fn}, of simple functions converging pointwise to f . Then∫

Yf (x,y)dλ = lim

n→∞

∫Y

fn(x,y)dλ ,

∫X

f (x,y)dµ = limn→∞

∫X

fn(x,y)dµ.

This follows from the monotone convergence theorem. Since

x→∫

Yfn(x,y)dλ