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.
[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).
(a real number product) denote its volume.
}(E)=\inf \left\{\sum _{k=1}^{\infty }\operatorname {vol} (C_{k}):{(C_{k})_{k\in \mathbb {N} }}{\text{ is a sequence of products of open intervals with }}E\subset \bigcup _{k=1}^{\infty }C_{k}\right\}.}
of the real numbers is reduced to its outer measure by coverage by sets of open intervals.
The total length of any covering interval set may overestimate the measure of
The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets.
Intuitively, it is the total length of those interval sets which fit
must not have some curious properties which causes a discrepancy in the measure of another set when
is used as a "mask" to "clip" that set, hinting at the existence of sets for which the Lebesgue outer measure does not give the Lebesgue measure.
The Lebesgue measure on Rn has the following properties: All the above may be succinctly summarized as follows (although the last two assertions are non-trivially linked to the following): 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.
If a subset of Rn has Hausdorff dimension less than n then it is a null set with respect to n-dimensional Lebesgue measure.
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.
The modern construction of the Lebesgue measure is an application of Carathéodory's extension theorem.
Fix n ∈ N. A box in Rn is a set of the form
For any subset A of Rn, we can define its outer measure λ*(A) by:
is a countable collection of boxes whose union covers
{\displaystyle \lambda ^{*}(A)=\inf \left\{\sum _{B\in {\mathcal {C}}}\operatorname {vol} (B):{\mathcal {C}}{\text{ is a countable collection of boxes whose union covers }}A\right\}.}
We then define the set A to be Lebesgue-measurable if for every subset S of Rn,
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.
The existence of sets that are not Lebesgue-measurable is a consequence of the set-theoretical axiom of choice, which is independent from many of the conventional systems of axioms for set theory.
The Vitali theorem, which follows from the axiom, states that there exist subsets of R that are not Lebesgue-measurable.
Assuming the axiom of choice, non-measurable sets with many surprising properties have been demonstrated, such as those of the Banach–Tarski paradox.
In 1970, Robert M. Solovay showed that the existence of sets that are not Lebesgue-measurable is not provable within the framework of Zermelo–Fraenkel set theory in the absence of the axiom of choice (see Solovay's model).
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).
It can be shown that there is no infinite-dimensional analogue of Lebesgue measure.
