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

334 CHAPTER 12. THE CONSTRUCTION OF MEASURESProof: 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 = f, off N, a set ofj-1 Fi having [ly X +++ X UL, measure zero and that f\ is a complex valued function mea-surable with respect to |J_, Fi. Suppose also that for some permutation of (1,2,--+ ,n)sis vJn)I ih, Jl JThenfel! ( i [ [FTi=l i=1Land the conclusion of Corollary 12.12.8 holds.Proof: Since | fi| is []/_, 4; measurable, it follows from Theorem 12.12.6 thathook, fildw;, ---dy,,[fila (i x tt)| \filax=E,)= | \flag@x=xE,).8VThus f € L! ([Ti_, Xi, 0) Fi, X--* X H,) 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,U) and (Y,.%,Vv) be o finite complete measure spaces andsuppose f >Ois ¥ x Y measurable. Then for a.e. x,y f(xy)is SY measurable. Similarly for a.e. y,x— f(x,y)is ¥ measurable.Proof: By Theorem 12.11.2, there exist x Y measurable functions, g and h and aset, N © F x SY of tt X A measure zero such that g < f < hand for (x,y) ¢ N, it followsthat g(x,y) =h(x,y). Then| [savan = [ navawx JY x JY