Measure, Integration & Real Analysis, 2021a

Section 5A Products of Measure Spaces 117
5A
Products of Measure Spaces
Products of σ-Algebras
Our first step in constructing product measures is to construct the product of two
σ-algebras. We begin with the following definition.
5.1 Definition rectangle
Suppose X and Y are sets. A rectangle in X × Y is a set of the form A × B,
where A ⊂ X and B ⊂ Y.
Keep the figure shown here in mind
when thinking of a rectangle in the sense
defined above. However, remember that
A and B need not be intervals as shown
in the figure. Indeed, the concept of an
interval makes no sense in the generality
of arbitrary sets.
Now we can define the product of two σ-algebras.
5.2 Definition product of two σ-algebras; S⊗T; measurable rectangle
Suppose (X, S) and (Y, T ) are measurable spaces. Then
• the product S⊗T is defined to be the smallest σ-algebra on X × Y that
contains
{A × B : A ∈S, B ∈T};
• a measurable rectangle in S⊗T is a set of the form A × B, where A ∈S
and B ∈T.
Using the terminology introduced in
the second bullet point above, we can say
that S⊗T is the smallest σ-algebra containing
all the measurable rectangles in
S⊗T. Exercise 1 in this section asks
you to show that the measurable rectangles
in S⊗T are the only rectangles in
X × Y that are in S⊗T.
The notation S×T is not used
because S and T are sets (of sets),
and thus the notation S×T
already is defined to mean the set of
all ordered pairs of the form (A, B),
where A ∈Sand B ∈T.
The notion of cross sections plays a crucial role in our development of product
measures. First, we define cross sections of sets, and then we define cross sections of
functions.
Measure, Integration & Real Analysis, by Sheldon Axler