Lebesgue measure

The Lebesgue measure is the standard notion of length, area, and volume in measure theory, extending the intuitive concepts of interval length and box volume to a vast class of subsets of Euclidean space. It is countably additive, translation-invariant, and assigns the expected measure to elementary geometric figures — the measure of an interval is its length, the measure of a rectangle is its area. Yet the Lebesgue measure is not universal: it cannot be defined for all subsets of the real line without contradiction. The Vitali set and other non-measurable sets lie outside its domain, and the attempt to extend the Lebesgue measure to all subsets — to create a universal