# Lebesgue measure

The**Lebesgue measure**is the standard way of assigning a 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, 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.

Table of contents |

2 Null sets 3 Construction of the Lebesgue measure 4 Relation to other measures 5 History |

### Properties

The Lebesgue measure has the following properties:

- If
*A*is a product of intervals of the form*I*_{1}x*I*_{2}x ... x*I*_{n}, then*A*is Lebesgue measurable and λ(*A*) = |*I*_{1}| · ... · |*I*_{n}|. Here, |*I*| denotes the length of the interval*I*as explained in the article on intervals. - 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 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*in*A*}, is also Lebesgue measurable and has the same measure as*A*.

- 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 λ([0, 1] x [0, 1] x ... x [0, 1]) = 1.

### 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, due to Carathéodory, proceeds as follows.

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.