Outer measure, measurable sets, and Lebesgue integration
Lebesgue measure extends "length" to highly irregular sets. The outer measure λ*(E) approximates from outside using countable covers by open intervals. A set E is (Lebesgue) measurable if for every set A: λ*(A) = λ*(A∩E) + λ*(A∩Eᶜ) (Carathéodory criterion). The Cantor set is a remarkable example: it has Lebesgue measure zero (removed intervals sum to 1) but is uncountable. The Vitali set (constructed using axiom of choice) is not Lebesgue measurable — it defies all consistent measure assignments.