Lebesgue measure

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 higher dimensional Euclidean n-spaces. For lower dimensions 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, hypervolume, or simply volume.^[1] 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).

Any closed [a, b] of real numbers is Lebesgue-measurable, and its Lebesgue measure is the length b − a. The open interval (a, b) has the same measure, since the difference between the two sets consists only of the end points a and b, which each have measure zero.

interval

Any of intervals [a, b] and [c, d] is Lebesgue-measurable, and its Lebesgue measure is (b − a)(d − c), the area of the corresponding rectangle.

Cartesian product

Moreover, every is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borel sets.^[5]^[6]

Borel set

Any set of real numbers has Lebesgue measure 0. In particular, the Lebesgue measure of the set of algebraic numbers is 0, even though the set is dense in $\mathbb {R}$ .

countable

The and the set of Liouville numbers are examples of uncountable sets that have Lebesgue measure 0.

Cantor set

If the holds then all sets of reals are Lebesgue-measurable. Determinacy is however not compatible with the axiom of choice.

axiom of determinacy

are examples of sets that are not measurable with respect to the Lebesgue measure. Their existence relies on the axiom of choice.

Vitali sets

are simple plane curves with positive Lebesgue measure^[7] (it can be obtained by small variation of the Peano curve construction). The dragon curve is another unusual example.

Osgood curves

Any line in $\mathbb {R} ^{n}$ , for $n\geq 2$ , has a zero Lebesgue measure. In general, every proper has a zero Lebesgue measure in its ambient space.

hyperplane

The can be calculated in terms of Euler's gamma function.

volume of an n-ball

Relation to other measures[edit]

The Borel measure agrees with the Lebesgue measure on those sets for which it is defined; however, there are many more Lebesgue-measurable sets than there are Borel measurable sets. The Borel measure is translation-invariant, but not complete.

The Haar measure can be defined on any locally compact group and is a generalization of the Lebesgue measure (Rⁿ with addition is a locally compact group).

The Hausdorff measure is a generalization of the Lebesgue measure that is useful for measuring the subsets of Rⁿ of lower dimensions than n, like submanifolds, for example, surfaces or curves in R³ and fractal sets. The Hausdorff measure is not to be confused with the notion of Hausdorff dimension.

It can be shown that there is no infinite-dimensional analogue of Lebesgue measure.

Lebesgue measure

interval

Cartesian product

Borel set

countable

Cantor set

axiom of determinacy

Vitali sets

Osgood curves

hyperplane

volume of an n-ball

Relation to other measures[edit]

4-volume

Edison Farah

Lebesgue's density theorem

Lebesgue measure of the set of Liouville numbers

Vitali set

Peano–Jordan measure