Measure, Integration & Real Analysis, 2021a

118 Chapter 5 Product Measures
5.3 Definition cross sections of sets; [E] a and [E] b
Suppose X and Y are sets and E ⊂ X × Y. Then for a ∈ X and b ∈ Y, the cross
sections [E] a and [E] b are defined by
[E] a = {y ∈ Y : (a, y) ∈ E} and [E] b = {x ∈ X : (x, b) ∈ E}.
5.4 Example cross sections of a subset of X × Y
5.5 Example cross sections of rectangles
Suppose X and Y are sets and A ⊂ X and B ⊂ Y. Ifa ∈ X and b ∈ Y, then
{
{
B if a ∈ A,
[A × B] a =
and [A × B] b A if b ∈ B,
=
∅ if a /∈ A
∅ if b /∈ B,
as you should verify.
The next result shows that cross sections preserve measurability.
5.6 cross sections of measurable sets are measurable
Suppose S is a σ-algebra on X and T is a σ-algebra on Y. IfE ∈S⊗T, then
[E] a ∈T for every a ∈ X and [E] b ∈Sfor every b ∈ Y.
Proof Let E denote the collection of subsets E of X × Y for which the conclusion
of this result holds. Then A × B ∈Efor all A ∈Sand all B ∈T (by Example 5.5).
The collection E is closed under complementation and countable unions because
and
[(X × Y) \ E] a = Y \ [E] a
[E 1 ∪ E 2 ∪···] a =[E 1 ] a ∪ [E 2 ] a ∪···
for all subsets E, E 1 , E 2 ,... of X × Y and all a ∈ X, as you should verify, with
similar statements holding for cross sections with respect to all b ∈ Y.
Because E is a σ-algebra containing all the measurable rectangles in S⊗T,we
conclude that E contains S⊗T.
Measure, Integration & Real Analysis, by Sheldon Axler