Measure, Integration & Real Analysis, 2021a

Section 2A Outer Measure on R 15
The definition of outer measure involves an infinite sum. The infinite sum ∑ ∞ k=1 t k
of a sequence t 1 , t 2 ,... of elements of [0, ∞] is defined to be ∞ if some t k = ∞.
Otherwise, ∑ ∞ k=1 t k is defined to be the limit (possibly ∞) of the increasing sequence
t 1 , t 1 + t 2 , t 1 + t 2 + t 3 ,...of partial sums; thus
∞
n
∑ t k = lim n→∞ ∑ t k .
k=1
k=1
2.3 Example finite sets have outer measure 0
Suppose A = {a 1 ,...,a n } is a finite set of real numbers. Suppose ε > 0. Define
a sequence I 1 , I 2 ,...of open intervals by
{
(ak − ε, a
I k =
k + ε) if k ≤ n,
∅ if k > n.
Then I 1 , I 2 ,... is a sequence of open intervals whose union contains A. Clearly
∑ ∞ k=1 l(I k)=2εn. Hence |A| ≤2εn. Because ε is an arbitrary positive number, this
implies that |A| = 0.
Good Properties of Outer Measure
Outer measure has several nice properties that are discussed in this subsection. We
begin with a result that improves upon the example above.
2.4 countable sets have outer measure 0
Every countable subset of R has outer measure 0.
Proof Suppose A = {a 1 , a 2 ,...} is a countable subset of R. Let ε > 0. Fork ∈ Z + ,
let
(
I k = a k − ε
2 k , a k + ε )
2 k .
Then I 1 , I 2 ,...is a sequence of open intervals whose union contains A. Because
∞
∑ l(I k )=2ε,
k=1
we have |A| ≤2ε. Because ε is an arbitrary positive number, this implies that
|A| = 0.
The result above, along with the result that the set Q of rational numbers is
countable, implies that Q has outer measure 0. We will soon show that there are far
fewer rational numbers than real numbers (see 2.17). Thus the equation |Q| = 0
indicates that outer measure has a good property that we want any reasonable notion
of size to possess.
Measure, Integration & Real Analysis, by Sheldon Axler