12.12. ANOTHER VERSION OF PRODUCT MEASURES 333

Theorem 12.12.7 Let {(Xi,Fi,µ i)}ni=1 be σ finite measure spaces and let ∏

ni=1 Fi denote

the smallest σ algebra which contains the measurable boxes of the form ∏ni=1 Ai where

Ai ∈Fi. Then there exists a measure, λ defined on ∏ni=1 Fi such that if f : ∏

ni=1 Xi→ [0,∞]

is ∏ni=1 Fi measurable, and (i1, · · · , in) is any permutation of (1, · · · ,n) , then∫

f dλ =∫

Xin

· · ·∫

Xi1

f dµ i1 · · ·dµ in

Let(

∏ni=1 Xi,∏

ni=1 Fi,λ

)denote the completion of this product measure space and let

f :n

∏i=1

Xi→ [0,∞]

be ∏ni=1 Fi measurable. Then there exists N ∈ ∏

ni=1 Fi such that λ (N) = 0 and a non-

negative function, f1 measurable with respect to ∏ni=1 Fi such that f1 = f off N and if

(i1, · · · , in) is any permutation of (1, · · · ,n) , then∫f dλ =

∫Xin

· · ·∫

Xi1

f1dµ i1 · · ·dµ in .

Furthermore, f1 may be chosen to satisfy either f1 ≤ f or f1 ≥ f .

Proof: This follows immediately from Theorem 12.12.6 and Theorem 12.11.2. By thesecond theorem, there exists a function f1 ≥ f such that f1 = f for all (x1, · · · ,xn) /∈ N, aset of ∏

ni=1 Fi having measure zero. Then by Theorem 12.11.1 and Theorem 12.12.6∫

f dλ =∫

f1dλ =∫

Xin

· · ·∫

Xi1

f1dµ i1 · · ·dµ in .

Since f1 = f off a set of measure zero, I will dispense with the subscript. Also it iscustomary to write

λ = µ1×·· ·×µn

andλ = µ1×·· ·×µn.

Thus in more standard notation, one writes∫f d (µ1×·· ·×µn) =

∫Xin

· · ·∫

Xi1

f dµ i1 · · ·dµ in

This theorem is often referred to as Fubini’s theorem. The next theorem is also called this.

Corollary 12.12.8 Suppose f ∈ L1(∏

ni=1 Xi,∏

ni=1 Fi,µ1×·· ·×µn

)where each Xi is a σ

finite measure space. Then if (i1, · · · , in) is any permutation of (1, · · · ,n) , it follows∫f d (µ1×·· ·×µn) =

∫Xin

· · ·∫

Xi1

f dµ i1 · · ·dµ in .