12.9. PRODUCT MEASURES 309

It follows y→∫

XQ (x,y)dµ is measurable and so by the monotone convergence theoremagain, ∫ ∫

XQ (x,y)dµdν =∫ ∞

∑i=1

XB′i(y)µ

(A′i)

dν

=∞

∑i=1

∫XB′i

(y)µ(A′i)

dν

=∞

∑i=1

ν(B′i)

µ(A′i). (12.9.34)

This shows the measurability conditions, 12.9.32 and 12.9.33 hold for Q ∈ R and alsoestablishes the formula for ρ (Q) , 12.9.34.

If ∪iAi×Bi and ∪ jC j×D j are two sets of R, then their intersection is

∪i∪ j (Ai∩C j)× (Bi∩D j)

a countable union of measurable rectangles. Thus finite intersections of sets of R are in R.This proves the lemma.

Now note that from the definition of R if you have a sequence of elements of R thentheir union is also in R. The next lemma will enable the definition of an outer measure.

Lemma 12.9.4 Suppose {Ri}∞

i=1 is a sequence of sets of R then

ρ (∪∞i=1Ri)≤

∞

∑i=1

ρ (Ri) .

Proof: Let Ri = ∪∞j=1Ai

j×Bij. Using Lemma 12.9.3, let {A′m×B′m}

∞

m=1 be a sequenceof disjoint rectangles each of which is contained in some Ai

j×Bij for some i, j such that

∪∞i=1Ri = ∪∞

m=1A′m×B′m.

Now defineSi ≡

{m : A′m×B′m ⊆ Ai

j×Bij for some j

}.

It is not important to consider whether some m might be in more than one Si. The importantthing to notice is that

∪m∈SiA′m×B′m ⊆ ∪∞

j=1Aij×Bi

j = Ri.

Then by Lemma 12.9.3,

ρ (∪∞i=1Ri) = ∑

mρ(A′m×B′m

)≤

∞

∑i=1

∑m∈Si

ρ(A′m×B′m

)≤

∞

∑i=1

ρ(∪m∈SiA

′m×B′m

)≤

∞

∑i=1

ρ (Ri) .