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. Often denoted λ ( ⋅ ) {\displaystyle \lambda (\cdot )} , 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. 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. One way to characterize the Lebesgue measure is to first define it on Borel sets, that is all sets that can be obtained by countably many operations of unions, set complement, and intersections, from the collection open intervals, so that it assigns the usual length to open intervals, and satisfies natural properties under taking limits of intervals. The Lebesgue measure can then be obtained by completing this Borel measure, by assigning zero measure to all subsets of Borel sets that already have zero measure. 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.
Elementary construction Lebesgue measure may be introduced as an extension of ordinary length, area, and volume. On the real line, the starting point is that an interval such as [ a , b ] {\displaystyle [a,b]} , ( a , b ) {\displaystyle (a,b)} , or [ a , b ) {\displaystyle [a,b)} should have length b − a {\displaystyle b-a} . In R n {\displaystyle \mathbb {R} ^{n}} , the corresponding elementary sets are rectangular boxes
B = I 1 × ⋯ × I n , {\displaystyle B=I_{1}\times \cdots \times I_{n},}
where each I i {\displaystyle I_{i}} is an interval. The volume of such a box is
vol ( B ) = ∏ i = 1 n | I i | . {\displaystyle \operatorname {vol} (B)=\prod _{i=1}^{n}|I_{i}|.}
A construction related to the Lebesgue measure is the Jordan content, which approximates regions by finite partitions into rectangular boxes. Similar to the Riemann integral, a set is Jordan measurable if there are such partitions that contain the region, and partitions contained within the region, that have an arbitrarily small difference between them. Jordan measure is not as robust as the Lebesgue measure, because there are fairly basic sets that are not Jordan measurable, such as the set of rational numbers. Lebesgue measure extends this assignment from intervals and boxes to a sufficiently large class of more complicated sets while preserving the basic rule of countable additivity: if E 1 , E 2 , … {\displaystyle E_{1},E_{2},\ldots } are pairwise disjoint measurable sets, then
λ ( ⋃ k = 1 ∞ E k ) = ∑ k = 1 ∞ λ ( E k ) . {\displaystyle \lambda \left(\bigcup _{k=1}^{\infty }E_{k}\right)=\sum _{k=1}^{\infty }\lambda (E_{k}).}
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