Measure, Integration & Real Analysis, 2021a

Section 5A Products of Measure Spaces 119
Now we define cross sections of functions.
5.7 Definition cross sections of functions; [ f ] a and [ f ] b
Suppose X and Y are sets and f : X × Y → R is a function. Then for a ∈ X and
b ∈ Y, the cross section functions [ f ] a : Y → R and [ f ] b : X → R are defined
by
[ f ] a (y) = f (a, y) for y ∈ Y and [ f ] b (x) = f (x, b) for x ∈ X.
5.8 Example cross sections
• Suppose f : R × R → R is defined by f (x, y) =5x 2 + y 3 . Then
[ f ] 2 (y) =20 + y 3 and [ f ] 3 (x) =5x 2 + 27
for all y ∈ R and all x ∈ R, as you should verify.
• Suppose X and Y are sets and A ⊂ X and B ⊂ Y. Ifa ∈ X and b ∈ Y, then
as you should verify.
[χ A × B
] a = χ A
(a)χ B
and [χ A × B
] b = χ B
(b)χ A
,
The next result shows that cross sections preserve measurability, this time in the
context of functions rather than sets.
5.9 cross sections of measurable functions are measurable
Suppose S is a σ-algebra on X and T is a σ-algebra on Y. Suppose
f : X × Y → R is an S⊗T-measurable function. Then
[ f ] a is a T -measurable function on Y for every a ∈ X
and
Proof
[ f ] b is an S-measurable function on X for every b ∈ Y.
Suppose D is a Borel subset of R and a ∈ X. Ify ∈ Y, then
y ∈ ([ f ] a ) −1 (D) ⇐⇒ [ f ] a (y) ∈ D
⇐⇒ f (a, y) ∈ D
⇐⇒ (a, y) ∈ f −1 (D)
⇐⇒ y ∈ [ f −1 (D)] a .
Thus
([ f ] a ) −1 (D) =[f −1 (D)] a .
Because f is an S⊗T-measurable function, f −1 (D) ∈S⊗T. Thus the equation
above and 5.6 imply that ([ f ] a ) −1 (D) ∈T. Hence [ f ] a is a T -measurable function.
The same ideas show that [ f ] b is an S-measurable function for every b ∈ Y.
Measure, Integration & Real Analysis, by Sheldon Axler