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Concept
Lebesgue measure
Summary
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n''-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by λ(A).
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 in 1902.
For any interval , or , in the set of real numbers, let denote its length. For any subset , the Lebesgue outer measure is defined as an infimum
The above definition can be generalised to higher dimensions as follows.
For any rectangular cuboid which is a product of open intervals, let denote its volume.
For any subset ,
Some sets satisfy the Carathéodory criterion, which requires that for every ,
The set of all such forms a σ-algebra. For any such , its Lebesgue measure is defined to be its Lebesgue outer measure: .
A set that does not satisfy the Carathéodory criterion is not Lebesgue-measurable. ZFC proves that non-measurable sets do exist; an example is the Vitali sets.
The first part of the definition states that the subset of the real numbers is reduced to its outer measure by coverage by sets of open intervals. Each of these sets of intervals covers in a sense, since the union of these intervals contains . The total length of any covering interval set may overestimate the measure of because is a subset of the union of the intervals, and so the intervals may include points which are not in . The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets.
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