# Lebesgue measure

In mathematics, the **Lebesgue measure**, is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. Sets which can be assigned a volume are called **Lebesgue measurable**; the volume or measure of the Lebesgue measurable set *A* is denoted by λ(*A*). A Lebesgue measure of ∞ is possible, but even so, assuming the axiom of choice, not all subsets of **R**^{n} are Lebesgue measurable. The "strange" behavior of non-measurable sets gives rise to such statements as the Banach-Tarski paradox, a consequence of the axiom of choice.

## Contents

## Examples

- If
*A*is an interval [*a*,*b*], then its Lebesgue measure is the length*b*−*a*. The open interval (*a*,*b*) has the same measure, since the difference between the two sets has measure zero. - If
*A*is the Cartesian product of intervals [*a*,*b*] and [*c*,*d*], then it is a rectangle and its Lebesgue measure is the area (*b*−*a*)(*d*−*c*). - The Cantor set is an example of uncountable set that has the Lebesgue measure of zero.

## Properties

The Lebesgue measure on **R**^{n} has the following properties:

- If
*A*is a cartesian product of intervals*I*_{1}×*I*_{2}× ... ×*I*_{n}, then*A*is Lebesgue measurable and Here, |*I*| denotes the length of the interval*I*. - If
*A*is a disjoint union of finitely many or countably many disjoint Lebesgue measurable sets, then*A*is itself Lebesgue measurable and λ(*A*) is equal to the sum (or infinite series) of the measures of the involved measurable sets. - If
*A*is Lebesgue measurable, then so is its complement. - λ(
*A*) ≥ 0 for every Lebesgue measurable set*A*. - If
*A*and*B*are Lebesgue measurable and*A*is a subset of*B*, then λ(*A*) ≤ λ(*B*). (A consequence of 2, 3 and 4.) - Countable unions and intersections of Lebesgue measurable sets are Lebesgue measurable. (A consequence of 2 and 3.)
- If
*A*is an open or closed subset of**R**^{n}(see metric space), then*A*is Lebesgue measurable. - If
*A*is a Lebesgue measurable set with λ(*A*) = 0 (a null set), then every subset of*A*is also a null set. - If
*A*is Lebesgue measurable and*x*is an element of**R**^{n}, then the*translation of A by x*, defined by*A*+*x*= {*a*+*x*:*a*∈*A*}, is also Lebesgue measurable and has the same measure as*A*.

All the above may be succinctly summarized as follows:

- The Lebesgue measurable sets form a σ-algebra containing all products of intervals, and λ is the unique complete translation-invariant measure on that σ-algebra with

The Lebesgue measure also has the property of being σ-finite.

## Null sets

A subset of **R**^{n} is a *null set* if, for every ε > 0, it can be covered with countably many products of *n* intervals whose total volume is at most ε. All countable sets are null sets, and so are sets in **R**^{n} whose dimension is smaller than *n*, for instance straight lines or circles in **R**^{2}.

In order to show that a given set *A* is Lebesgue measurable, one usually tries to find a "nicer" set *B* which differs from *A* only by a null set (in the sense that the symmetric difference (*A* − *B*) ∪ (*B* − *A*) is a null set) and then shows that *B* can be generated using countable unions and intersections from open or closed sets.

## Construction of the Lebesgue measure

The modern construction of the Lebesgue measure, based on outer measures, is due to Carathéodory. It proceeds as follows:

For *any* subset *B* of **R**^{n}, we can define

Here, vol(*M*) is sum of the product of the lengths of the involved intervals. We then define the set *A* to be Lebesgue measurable if

for all sets *B*. These Lebesgue measurable sets form a σ-algebra, and the Lebesgue measure is defined by λ(*A*) = λ^{*}(*A*) for any Lebesgue measurable set *A*.

According to the Vitali theorem there exists a subset of the real numbers **R** that is not Lebesgue measurable.

## Relation to other measures

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 (**R**^{n} with addition is a locally compact group).

The Hausdorff measure (see Hausdorff dimension) is a generalization of the Lebesgue measure that is useful for measuring the sets of **R**^{n} of lower dimensions than *n*, like submanifolds, for example, surfaces or curves in **R**^{3} and fractal sets.

## History

Henri Lebesgue described his measure in 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.

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