12.12. ANOTHER VERSION OF PRODUCT MEASURES 331

Then K ⊆ G . This is obvious.Next I want to show that if E ∈ G then EC ∈ G . Observe XEC = 1−XE and so∫

Y

∫X

XEC dµdν =∫

Y

∫X(1−XE)dµdν

=∫

X

∫Y(1−XE)dνdµ

=∫

X

∫Y

XEC dνdµ

which shows that if E ∈ G , then EC ∈ G .Next I want to show G is closed under countable unions of disjoint sets of G . Let {Ai}

be a sequence of disjoint sets from G . Then∫Y

∫X

X∪∞i=1Aidµdν =

∫Y

∫X

∑i=1

XAidµdν

=∫

Y

∑i=1

∫X

XAidµdν

=∞

∑i=1

∫Y

∫X

XAidµdν

=∞

∑i=1

∫X

∫Y

XAidνdµ

=∫

X

∑i=1

∫Y

XAidνdµ

=∫

X

∫Y

∑i=1

XAidνdµ

=∫

X

∫Y

X∪∞i=1Aidνdµ, (12.12.56)

the interchanges between the summation and the integral depending on the monotone con-vergence theorem. Thus G is closed with respect to countable disjoint unions.

From Lemma 12.12.3, G ⊇ σ (K ) . Also the computation in 12.12.56 implies that onσ (K ) one can define a measure, denoted by µ×ν and that for every E ∈ σ (K ) ,

(µ×ν)(E) =∫

Y

∫X

XEdµdν =∫

X

∫Y

XEdνdµ. (12.12.57)

Now here is Fubini’s theorem.

Theorem 12.12.4 Let f : X ×Y → [0,∞] be measurable with respect to the σ algebra,σ (K ) just defined and let µ ×ν be the product measure of 12.12.57 where µ and ν arefinite measures on (X ,F ) and (Y,S ) respectively. Then∫

X×Yf d (µ×ν) =

∫Y

∫X

f dµdν =∫

X

∫Y

f dνdµ.

12.12. ANOTHER VERSION OF PRODUCT MEASURES 331Then #% CG. This is obvious.Next I want to show that if E € Y then E© € Y. Observe 2c = 1— Ze and so| |, %ecauav [ [= %e)auav| [0-2avau| | Xp cdvdxX JYwhich shows that if E € Y, then EC € Y.Next I want to show ¥ is closed under countable unions of disjoint sets of Y. Let {A;}be a sequence of disjoint sets from Y. Then| | Boe adudv = | [ Y 24,dudvYIX Y JX j=[¥ | PaauavJy & Ixr/ 2Xy,dudvi=] /¥ YXy | f avei=1/X VY[X [ GavanJx & Jy| [YX PaavanXY j=]J [ Mr mavdn, (12.12.56)X JY =the interchanges between the summation and the integral depending on the monotone con-vergence theorem. Thus Y is closed with respect to countable disjoint unions.From Lemma 12.12.3, 9 D o (.#). Also the computation in 12.12.56 implies that ono (.#) one can define a measure, denoted by uw x v and that for every E€ 0 (.%),(uxvy(e)= ff Heapdy= | | %edvap. (12.12.57)y JX x JyNow here is Fubini’s theorem.Theorem 12.12.4 Let f : X x Y — [0,0] be measurable with respect to the o algebra,0 (.#) just defined and let LU x Vv be the product measure of 12.12.57 where and Vv arefinite measures on (X,#) and (Y,./) respectively. Then|. fatuxv)= [| fauav= [| favap.