306 CHAPTER 10. THE ABSTRACT LEBESGUE INTEGRAL

and so the definition with respect to the two different increasing sequences gives the samething. Thus the definition is well defined. (µ×ν) is a measure because if the Ei are disjointE ×F measurable sets and E = ∪iEi,

(µ×ν)(E)≡∑n

∑m(µn×νm)(∪iEi∩ (Xn×Ym)) = ∑

n∑m

∑i(µn×νm)(Ei∩ (Xn×Ym))

= ∑i

∑n

∑m(µn×νm)(Ei∩ (Xn×Ym))≡∑

i(µ×ν)(Ei) ■

Theorem 10.14.8 Let f : X ×Y → [0,∞] be measurable with respect to the σ al-gebra, σ (K ) just defined as the smallest σ algebra containing the measurable rectangles,and let µ × ν be the product measure of 10.27 where µ and ν are σ finite measures on(X ,E ) and (Y,F ) respectively. (10.14.1) Then∫

X×Yf d (µ×ν) =

∫Y

∫X

f dµdν =∫

X

∫Y

f dνdµ. (10.28)

Proof: Letting E ∈ E ×F ,∫X×Y

XEd (µ×ν)≡ (µ×ν)(E)≡∑n

∑m(µn×νm)(E ∩ (Xn×Ym))

= ∑n

∑m

∫Yn

∫Xn

XEdµndνn =∫

Y

∫X

XEdµdν

the last coming from a use of the monotone convergence theorem applied to sums. Itfollows that 10.28 holds for simple functions and then from monotone convergence theoremand Theorem 9.1.6, it holds for nonnegative E ×F measurable functions. ■

It is also useful to note that all the above holds for ∏pi=1 Xi in place of X ×Y and µ i a

measure on Ei a σ algebra of sets of Xi. You would simply modify the definition of G in10.25 including all permutations for the iterated integrals and for K you would use sets ofthe form ∏

pi=1 Ai where Ai is measurable. Everything goes through exactly as above.

Thus the following is mostly obtained.

Theorem 10.14.9 Let {(Xi,Ei,µ i)}pi=1 be σ finite measure spaces and ∏

pi=1 Ei de-

notes the smallest σ algebra which contains the measurable boxes of the form ∏pi=1 Ai

where Ai ∈ Ei. Then there exists a measure λ defined on ∏pi=1 Ei such that if f : ∏

pi=1 Xi

→ [0,∞] is ∏pi=1 Ei measurable, (i1, · · · , ip) is any permutation of (1, · · · , p) , then∫

f dλ =∫

Xip

· · ·∫

Xi1

f dµ i1 · · ·dµ ip(10.29)

If each Xi is a complete separable metric space such that µ i is finite on balls and Ei containsB (Xi), the Borel sets of Xi, then λ is a regular measure on a σ algebra of sets of ∏

pi=1 Xi

with the metric given by d (x,y) ≡ max{d (xi,yi) : xi,yi ∈ Xi}, which includes the Borelsets.

Proof: It remains to verify the last claim. This is because all sets ∏pi=1 B(ξ i,r) are

contained in ∏pi=1 Ei and are the open balls for the topology of ∏

pi=1 Xi. Then by separability

of each Xi, the product ∏pi=1 Xi is also separable and so this product with the above metric

is completely separable. Thus every open set is the countable union of these sets so opensets are in ∏

pi=1 Ei which consequently contains the Borel sets. Now from Corollary 9.8.9,

λ is regular because it is finite on balls. ■The conclusion 10.29 is called Fubini’s theorem. More generally