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