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.]