Measure, Integration & Real Analysis, 2021a

Section 2A Outer Measure on R 21
The previous result has the following important corollary. You may be familiar
with Georg Cantor’s (1845–1918) original proof of the next result. The proof using
outer measure that is presented here gives an interesting alternative to Cantor’s proof.
2.17 nontrivial intervals are uncountable
Every interval in R that contains at least two distinct elements is uncountable.
Proof
Suppose I is an interval that contains a, b ∈ R with a < b. Then
|I| ≥|[a, b]| = b − a > 0,
where the first inequality above holds because outer measure preserves order (see 2.5)
and the equality above comes from 2.14. Because every countable subset of R has
outer measure 0 (see 2.4), we can conclude that I is uncountable.
Outer Measure is Not Additive
We have had several results giving nice
properties of outer measure. Now we
come to an unpleasant property of outer
measure.
If outer measure were a perfect way to
assign a size as an extension of the lengths
of intervals, then the outer measure of the
union of two disjoint sets would equal the
Outer measure led to the proof
above that R is uncountable. This
application of outer measure to
prove a result that seems
unconnected with outer measure is
an indication that outer measure has
serious mathematical value.
sum of the outer measures of the two sets. Sadly, the next result states that outer
measure does not have this property.
In the next section, we begin the process of getting around the next result, which
will lead us to measure theory.
2.18 nonadditivity of outer measure
There exist disjoint subsets A and B of R such that
|A ∪ B| ̸= |A| + |B|.
Proof For a ∈ [−1, 1], let ã be the set of numbers in [−1, 1] that differ from a by a
rational number. In other words,
If a, b ∈ [−1, 1] and ã ∩ ˜b ̸= ∅, then
ã = ˜b. (Proof: Suppose there exists d ∈
ã ∩ ˜b. Then a − d and b − d are rational
numbers; subtracting, we conclude that
a − b is a rational number. The equation
ã = {c ∈ [−1, 1] : a − c ∈ Q}.
Think of ã as the equivalence class
of a under the equivalence relation
that declares a, c ∈ [−1, 1] to be
equivalent if a − c ∈ Q.
a − c =(a − b)+(b − c) now implies that if c ∈ [−1, 1], then a − c is a rational
number if and only if b − c is a rational number. In other words, ã = ˜b.)
Measure, Integration & Real Analysis, by Sheldon Axler