334 CHAPTER 12. THE CONSTRUCTION OF MEASURES

Proof: Just apply Theorem 12.12.7 to the positive and negative parts of the real andimaginary parts of f . This proves the theorem.

Here is another easy corollary.

Corollary 12.12.9 Suppose in the situation of Corollary 12.12.8, f = f1 off N, a set of∏

ni=1 Fi having µ1×·· ·×µn measure zero and that f1 is a complex valued function mea-

surable with respect to ∏ni=1 Fi. Suppose also that for some permutation of (1,2, · · · ,n)

,( j1, · · · , jn) ∫X jn

· · ·∫

X j1

| f1|dµ j1 · · ·dµ jn < ∞.

Then

f ∈ L1

(n

∏i=1

Xi,n

∏i=1

Fi,µ1×·· ·×µn

)and the conclusion of Corollary 12.12.8 holds.

Proof: Since | f1| is ∏ni=1 Fi measurable, it follows from Theorem 12.12.6 that

∞ >∫

X jn

· · ·∫

X j1

| f1|dµ j1 · · ·dµ jn

=∫| f1|d (µ1×·· ·×µn)

=∫| f1|d (µ1×·· ·×µn)

=∫| f |d (µ1×·· ·×µn) .

Thus f ∈ L1(∏

ni=1 Xi,∏

ni=1 Fi,µ1×·· ·×µn

)as claimed and the rest follows from Corol-

lary 12.12.8. This proves the corollary.The following lemma is also useful.

Lemma 12.12.10 Let (X ,F ,µ) and (Y,S ,ν) be σ finite complete measure spaces andsuppose f ≥ 0 is F ×S measurable. Then for a.e. x,

y→ f (x,y)

is S measurable. Similarly for a.e. y,

x→ f (x,y)

is F measurable.

Proof: By Theorem 12.11.2, there exist F ×S measurable functions, g and h and aset, N ∈F ×S of µ ×λ measure zero such that g ≤ f ≤ h and for (x,y) /∈ N, it followsthat g(x,y) = h(x,y) . Then ∫

X

∫Y

gdνdµ =∫

X

∫Y

hdνdµ