Measure, Integration & Real Analysis, 2021a

124 Chapter 5 Product Measures
The next result will allow us to define the product of two σ-finite measures.
5.20 measure of cross section is a measurable function
Suppose (X, S, μ) and (Y, T , ν) are σ-finite measure spaces. If E ∈S⊗T,
then
(a) x ↦→ ν([E] x ) is an S-measurable function on X;
(b) y ↦→ μ([E] y ) is a T -measurable function on Y.
Proof We will prove (a). If E ∈S⊗T, then [E] x ∈T for every x ∈ X (by 5.6);
thus the function x ↦→ ν([E] x ) is well defined on X.
We first consider the case where ν is a finite measure. Let
M = {E ∈S⊗T : x ↦→ ν([E] x ) is an S-measurable function on X}.
We need to prove that M = S⊗T.
If A ∈Sand B ∈T, then ν([A × B] x )=ν(B)χ A
(x) for every x ∈ X (by
Example 5.5). Thus the function x ↦→ ν([A × B] x ) equals the function ν(B)χ A
(as
a function on X), which is an S-measurable function on X. Hence M contains all
the measurable rectangles in S⊗T.
Let A denote the set of finite unions of measurable rectangles in S⊗T. Suppose
E ∈A. Then by 5.13(b), E is a union of disjoint measurable rectangles E 1 ,...,E n .
Thus
ν([E] x )=ν([E 1 ∪···∪E n ] x )
= ν([E 1 ] x ∪···∪[E n ] x )
= ν([E 1 ] x )+···+ ν([E n ] x ),
where the last equality holds because ν is a measure and [E 1 ] x ,...,[E n ] x are disjoint.
The equation above, when combined with the conclusion of the previous paragraph,
shows that x ↦→ ν([E] x ) is a finite sum of S-measurable functions and thus is an
S-measurable function. Hence E ∈M. We have now shown that A⊂M.
Our next goal is to show that M is a monotone class on X × Y. To do this, first
suppose E 1 ⊂ E 2 ⊂··· is an increasing sequence of sets in M. Then
(
ν [
∞⋃
k=1
) ( ⋃ ∞
E k ] x = ν
k=1
)
([E k ] x )
= lim
k→∞
ν([E k ] x ),
where we have used 2.59. Because the pointwise limit of S-measurable functions
is S-measurable (by 2.48), the equation above shows that x ↦→ ν ( [ ⋃ ∞
k=1
E k ] x
)
is
an S-measurable function. Hence ⋃ ∞
k=1
E k ∈M. We have now shown that M is
closed under countable increasing unions.
Measure, Integration & Real Analysis, by Sheldon Axler