Measure, Integration & Real Analysis, 2021a

Section 5C Lebesgue Integration on R n 137
5.36 product of open sets is open
Suppose G 1 is an open subset of R m and G 2 is an open subset of R n . Then
G 1 × G 2 is an open subset of R m+n .
Proof Suppose (x, y) ∈ G 1 × G 2 . Then there exists an open cube D in R m centered
at x and an open cube E in R n centered at y such that D ⊂ G 1 and E ⊂ G 2 .By
reducing the size of either D or E, we can assume that the cubes D and E have the
same side length. Thus D × E is an open cube in R m+n centered at (x, y) that is
contained in G 1 × G 2 .
We have shown that an arbitrary point in G 1 × G 2 is the center of an open cube
contained in G 1 × G 2 . Hence G 1 × G 2 is an open subset of R m+n .
When n = 1, the definition below of a Borel subset of R 1 agrees with our previous
definition (2.29) of a Borel subset of R.
5.37 Definition Borel set; B n
• A Borel subset of R n is an element of the smallest σ-algebra on R n containing
all open subsets of R n .
• The σ-algebra of Borel subsets of R n is denoted by B n .
Recall that a subset of R is open if and only if it is a countable disjoint union of
open intervals. Part (a) in the result below provides a similar result in R n , although
we must give up the disjoint aspect.
5.38 open sets are countable unions of open cubes
(a) A subset of R n is open in R n if and only if it is a countable union of open
cubes in R n .
(b) B n is the smallest σ-algebra on R n containing all the open cubes in R n .
Proof We will prove (a), which clearly implies (b).
The proof that a countable union of open cubes is open is left as an exercise for
the reader (actually, arbitrary unions of open cubes are open).
To prove the other direction, suppose G is an open subset of R n . For each x ∈ G,
there is an open cube centered at x that is contained in G. Thus there is a smaller
cube C x such that x ∈ C x ⊂ G and all coordinates of the center of C x are rational
numbers and the side length of C x is a rational number. Now
G = ⋃
C x .
x∈G
However, there are only countably many distinct cubes whose center has all rational
coordinates and whose side length is rational. Thus G is the countable union of open
cubes.
Measure, Integration & Real Analysis, by Sheldon Axler