Measure, Integration & Real Analysis, 2021a

Section 2D Lebesgue Measure 51
The tools we have constructed now allow us to prove that outer measure, when
restricted to the Borel sets, is a measure.
2.68 outer measure is a measure on Borel sets
Outer measure is a measure on (R, B), where B is the σ-algebra of Borel subsets
of R.
Proof Suppose B 1 , B 2 ,...is a disjoint sequence of Borel subsets of R. Then for
each n ∈ Z + we have ∣ ∣∣ ⋃ ∞ ∣ ∣ ∣∣ ∣∣ ⋃ n ∣ ∣∣
B k ≥ B k
k=1
=
k=1
n
∑
k=1
|B k |,
where the first line above follows from 2.5 and the last line follows from 2.66 (and
induction on n). Taking the limit as n → ∞, wehave ∣ ⋃ ∣
∞
k=1
B k ≥ ∑ ∞ k=1 |B k|.
The inequality in the other directions follows from countable subadditivity of outer
measure (2.8). Hence
∞⋃ ∣ ∞ ∣∣
∣ B k = ∑ |B k |.
k=1 k=1
Thus outer measure is a measure on the σ-algebra of Borel subsets of R.
The result above implies that the next definition makes sense.
2.69 Definition Lebesgue measure
Lebesgue measure is the measure on (R, B), where B is the σ-algebra of Borel
subsets of R, that assigns to each Borel set its outer measure.
In other words, the Lebesgue measure of a set is the same as its outer measure,
except that the term Lebesgue measure should not be applied to arbitrary sets but
only to Borel sets (and also to what are called Lebesgue measurable sets, as we will
soon see). Unlike outer measure, Lebesgue measure is actually a measure, as shown
in 2.68. Lebesgue measure is named in honor of its inventor, Henri Lebesgue.
The cathedral in Beauvais, the
French city where Henri
Lebesgue (1875–1941) was
born. Much of what we call
Lebesgue measure and
Lebesgue integration was
developed by Lebesgue in his
1902 PhD thesis. Émile Borel
was Lebesgue’s PhD thesis
advisor. CC-BY-SA James Mitchell
Measure, Integration & Real Analysis, by Sheldon Axler