How we defined Lebesgue measure on R
Definition. Let
I = {(a, b] : a, b ∈ R, a ≤ b}
and define
m : I → [0, ∞) m (a, b] = b − a.
S
P∞
Lemma 1. Let I ∈ I. If I1 , I2 , . . . ∈ I with I ⊆ ∞
j=1 Ij , then m(I) ≤
j=1 m(Ij ).
Definition. If E ⊆ R, the Lebesgue outer measure of E is
(∞
)
∞
X
[
m∗ (E) = inf
m(Ij ) : I1 , I2 , . . . ∈ I and E ⊆
Ij .
j=1
Proposition 2 (Properties of m∗ ).
j=1
(a) m∗ (∅) = 0
(b) m∗ is increasing
(c) m∗ is countably subadditive
(d) m∗ extends m [This uses Lemma 1]
(e) m∗ is translation-invariant [basically because m is]
Definition. A set E ⊆ R is Lebesgue measurable if, for every set A ⊆ R, we have
m∗ (A) = m∗ (A ∩ E) + m∗ (A ∩ E c ).
We write L = {E ⊆ R : E is Lebesgue measurable}.
Lemma 3. If I, J ∈ I then m∗ (J) = m∗ (J ∩ I) + m∗ (J ∩ I c ).
Theorem 4. I ⊆ L.
[This follows from Lemma 3]
Definition. Lebesgue measure on R is the map µ : L → [0, ∞] given by µ = m∗ |L . That
is,
µ(A) = m∗ (A) for A ∈ L.
Remark. µ inherits all of the properties of m∗ from Proposition 2. In particular, µ
extends m and µ is translation-invariant.
Theorem 5.
(a) L is a σ-algebra of subsets of R
(b) µ : L → [0, ∞] is a measure.
[µ is countably additive, whereas m∗ is only countably subadditive.]