Lebesgue measure

In mathematics, Lebesgue measure is the standard way of assigning a notion of length to subsets of the real line, area to regions of the Euclidean plane, and volume to subsets of Euclidean space in dimensions three and higher. It is used throughout mathematical analysis, especially in the definition of the Lebesgue integral and in statements that hold "almost everywhere," meaning except on a set whose Lebesgue measure is zero. Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation Intégrale, Longueur, Aire in 1902.

Lebesgue measure extends ordinary geometric length (or volume) in a way that is compatible with countable unions and other kinds of countable limits of sets. For example, every countable subset of the real line has Lebesgue measure zero, being a countable union of points, which have no length, while many uncountable sets also have measure zero. The measure is not defined on every subset of the real line (or Euclidean space) under the usual axioms of set theory: the sets to which it applies are called Lebesgue-measurable.