In mathematics, the Lebesgue measure, named after Henri Lebesgue, 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 Rn 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.
Lebesgue measure is often denoted , but this should not be confused with the distinct notion of a volume form.
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The Lebesgue measure on Rn has the following properties:
All the above may be succinctly summarized as follows:
The Lebesgue measure also has the property of being σ-finite.
A subset of Rn 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.
If a subset of Rn has Hausdorff dimension less than n then it is a null set with respect to n-dimensional Lebesgue measure. Here Hausdorff dimension is relative to the Euclidean metric on Rn (or any metric Lipschitz equivalent to it). On the other hand a set may have topological dimension less than n and have positive n-dimensional Lebesgue measure. An example of this is the Smith-Volterra-Cantor set which has topological dimension 0 yet has positive 1-dimensional Lebesgue measure.
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 show that B can be generated using countable unions and intersections from open or closed sets.
The modern construction of the Lebesgue measure, based on outer measures, is due to Carathéodory. It proceeds as follows.
Fix . A box in
is a set of the form
, where
. The volume
of this box is defined to be
For any subset A of Rn, we can define its outer measure by:
We then define the set A to be Lebesgue measurable if
for all sets . 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. Much more is true: if A is any subset of of positive measure, then A has subsets which are not Lebesgue measurable.
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 (see Hausdorff dimension) 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.
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.[1]