Measure, Integration & Real Analysis, 2021a

138 Chapter 5 Product Measures
The next result tells us that the collection of Borel sets from various dimensions
fit together nicely.
5.39 product of the Borel subsets of R m and the Borel subsets of R n
B m ⊗B n = B m+n .
Proof Suppose E is an open cube in R m+n . Thus E is the product of an open cube
in R m and an open cube in R n . Hence E ∈B m ⊗B n . Thus the smallest σ-algebra
containing all the open cubes in R m+n is contained in B m ⊗B n .Now5.38(b) implies
that B m+n ⊂B m ⊗B n .
To prove the set inclusion in the other direction, temporarily fix an open set G
in R n . Let
E = {A ⊂ R m : A × G ∈B m+n }.
Then E contains every open subset of R m (as follows from 5.36). Also, E is closed
under countable unions because
( ⋃ ∞ ) ⋃ ∞
A k × G = (A k × G).
k=1
Furthermore, E is closed under complementation because
(
R m \ A ) (
)
× G = (R m × R n ) \ (A × G) ∩ ( R m × G ) .
Thus E is a σ-algebra on R m that contains all open subsets of R m , which implies that
B m ⊂E. In other words, we have proved that if A ∈B m and G is an open subset of
R n , then A × G ∈B m+n .
Now temporarily fix a Borel subset A of R m . Let
k=1
F = {B ⊂ R n : A × B ∈B m+n }.
The conclusion of the previous paragraph shows that F contains every open subset of
R n . As in the previous paragraph, we also see that F is a σ-algebra. Hence B n ⊂F.
In other words, we have proved that if A ∈B m and B ∈B n , then A × B ∈B m+n .
Thus B m ⊗B n ⊂B m+n , completing the proof.
The previous result implies a nice associative property. Specifically, if m, n, and
p are positive integers, then two applications of 5.39 give
(B m ⊗B n ) ⊗B p = B m+n ⊗B p = B m+n+p .
Similarly, two more applications of 5.39 give
B m ⊗ (B n ⊗B p )=B m ⊗B n+p = B m+n+p .
Thus (B m ⊗B n ) ⊗B p = B m ⊗ (B n ⊗B p ); hence we can dispense with parentheses
when taking products of more than two Borel σ-algebras. More generally, we could
have defined B m ⊗B n ⊗B p directly as the smallest σ-algebra on R m+n+p containing
{A × B × C : A ∈B m , B ∈B n , C ∈B p } and obtained the same σ-algebra (see
Exercise 3 in this section).
Measure, Integration & Real Analysis, by Sheldon Axler