308 CHAPTER 10. THE ABSTRACT LEBESGUE INTEGRAL

That is hF does not depend on E ∈ E . Note also that 10.30 shows right away that hF (x)≤ 1a.e. Just let F = Y . Also, this shows that hY (x) = 1 for α a.e. x because from 10.30,∫

X×YXE (x)dµ = µ (E×Y ) = α (E) =

∫X

XE (x)hY (x)dα

where hY (x)≤ 1. Now let Em =[hY < 1− 1

m

]so α (Em)≤

(1− 1

m

)α (Em) Then the above

shows α (Em) = 0 and so, taking a union for m ∈N, yields that the set where hY is less than1 has α measure zero.

Now F →∫

X XE (x)hF (x)dα is clearly a measure because∫

X×Y XE (x)XF (y)dµ =∫X XE (x)hF (x)dα implies that if {Fi} are disjoint, then h∪iFi = ∑i hFi this by the unique-

ness in the Radon Nikodym theorem. That is, for fixed x,F→ hF (x) is a measure νx. SincehY (x) = 1 for α a.e. x and 0≤ hF (x)≤ 1,hY (x) = 1 α a.e., we can let νx be a probabilitymeasure for α a.e. x. Summarizing,∫

X×YXE (x)XF (y)dµ =

∫X

XE (x)∫

YXF (y)dνxdα =

∫X

∫Y

XE×F (x,y)dνxdα

If ν̂x also works, then it must equal νx for α a.e. x.Let the π system K consist of E×F where E ∈ E and F ∈F . Let G be those sets

A of E ×F ≡ σ (K ) such that∫

X×Y XAdµ =∫

X∫

Y XA (x,y)dνxdα Then G contains Kand is closed with respect to countable disjoint unions and complements, the latter comingfrom the observation that X ×Y ∈ K which allows the same kind of argument used inthe above treatment of product measures. Therefore, by Dynkin’s lemma, G = σ (K ) andso, using approximation with simple functions and the monotone convergence theorem, weobtain that for any f which is E ×F ≡ σ (K ) measurable and nonnegative the iteratedintegrals make sense and

∫X×Y f dµ =

∫X∫

Y f dνxdα ■These measures νx are called slicing measures. They can be used to define what is

meant by independent random variables in probability. This also shows that a given µ is aproduct measure exactly when the νx don’t depend on x.

Consider now many spaces ∏ni=1 Xi where µ is a measure on E ≡ ∏

ni=1 Ei where this

denotes the product measurable sets from the (Xi,Ei) . Then for f nonnegative and E mea-surable, ∫

X1×···×Xn

f dµ =∫

X1×···×Xn−1

∫Xn

f dν(x1,··· ,xn−1) (xn)dν (x1, · · · ,xn−1)

Here for E ∈∏n−1i=1 Ei, ν (E)≡ ν (E×Xn) . This ν is denoted as ν (x1, · · · ,xn−1). Then this

equals ∫X1×···×Xn−2

∫Xn−1

∫Xn

f dν(x1,··· ,xn−1) (xn)dν(x1,··· ,xn−2) (xn−1)dν (x1, · · · ,xn−2)

...∫X1

· · ·∫

Xn

f dν(x1,··· ,xn−1) (xn)dν(x1,··· ,xn−2) (xn−1) · · ·dνx1 (x2)dν (x1)

where for E ∈ E1

Corollary 10.14.13 For E ≡ ∏ni=1 Ei, and f : ∏

ni=1 Xi → [0,∞] ,for (Xi,Ei) a measur-

able space and for µ a finite probability measure on E , meaning µ (∏ni=1 Xi) = 1, there are

probability measures as denoted below by ν with various subscripts such that∫X1×···×Xn

f dµ =∫

X1

· · ·∫

Xn

f dν(x1,··· ,xn−1) (xn)dν(x1,··· ,xn−2) (xn−1) · · ·dνx1 (x2)dν (x1)