9.12. ITERATED INTEGRALS 223

= ∑i

∑n

∑m(µn×νm)(Ei∩ (Xn×Ym))≡∑

i(µ×ν)(Ei) ■

Theorem 9.12.8 Let f : X ×Y → [0,∞] be measurable with respect to the σ alge-bra, σ (K ) just defined as the smallest σ algebra containing the measurable rectangles,and let µ×ν be the product measure of 9.17 where µ and ν are σ finite measures on (X ,E )and (Y,F ) respectively. (9.12.1) Then∫

X×Yf d (µ×ν) =

∫Y

∫X

f dµdν =∫

X

∫Y

f dνdµ. (9.18)

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 9.18 holds for simple functions and then from monotone convergence theoremand Theorem 8.1.6, 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 in9.15 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 9.12.9 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 a σ algebra ∏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(9.19)

The conclusion 9.19 is called Fubini’s theorem. More generally

Theorem 9.12.10 Suppose, in the situation of Theorem 9.12.9 f ∈ L1 with respectto the measure λ . Then 9.19 continues to hold.

Proof: It suffices to prove this for f having real values because if this is shown thegeneral case is obtained by taking real and imaginary parts. Since f ∈ L1

(∏

pi=1 Xi

),∫

| f |dλ < ∞ and so both 12 (| f |+ f ) and 1

2 (| f |− f ) are in L1(∏

pi=1 Xi

)and are each non-

negative. Hence from Theorem 9.12.9,∫f dλ =

∫ [12(| f |+ f )− 1

2(| f |− f )

]dλ =

∫ 12(| f |+ f )dλ −

∫ 12(| f |− f )dλ

=∫· · ·∫ 1

2(| f |+ f )dµ i1 · · ·dµ ip

−∫· · ·∫ 1

2(| f |− f )dµ i1 · · ·dµ ip

=∫· · ·∫ 1

2(| f |+ f )− 1

2(| f |− f )dµ i1 · · ·dµ ip

=∫· · ·∫

f dµ i1 · · ·dµ ip■

The following corollary is a convenient way to verify the hypotheses of the above theorem.