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 dimensionalEuclidean 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).
Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the
Lebesgue integral. Both were published as part of his dissertation in 1902.[2]
Definition
For any
interval, or , in the set of real numbers, let denote its length. For any subset , the Lebesgue
outer measure[3] is defined as an
infimum
The above definition can be generalised to higher dimensions as follows.[4]
For any
rectangular cuboid which is a
product of open intervals, let denote its volume.
For any subset ,
The sets that satisfy the
Carathéodory criterion are said to be Lebesgue-measurable, with its Lebesgue measure being defined as its Lebesgue outer measure: . The set of all such forms a
σ-algebra.
The first part of the definition states that the subset of the real numbers is reduced to its outer measure by coverage by sets of open intervals. Each of these sets of intervals covers in a sense, since the union of these intervals contains . The total length of any covering interval set may overestimate the measure of because is a subset of the union of the intervals, and so the intervals may include points which are not in . 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 most tightly and do not overlap.
That characterizes the Lebesgue outer measure. Whether this outer measure translates to the Lebesgue measure proper depends on an additional condition. This condition is tested by taking subsets of the real numbers using as an instrument to split into two partitions: the part of which intersects with and the remaining part of which is not in : the set difference of and . These partitions of are subject to the outer measure. If for all possible such subsets of the real numbers, the partitions of cut apart by have outer measures whose sum is the outer measure of , then the outer Lebesgue measure of gives its Lebesgue measure. Intuitively, this condition means that the set 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. (Such sets are, in fact, not Lebesgue-measurable.)
Examples
Any closed
intervala, 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.
Any
Cartesian product 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.
Moreover, every
Borel set is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borel sets.[5][6]
Any
countable 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 .
If A is a
disjoint union of
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.)
Countable
unions and
intersections of Lebesgue-measurable sets are Lebesgue-measurable. (Not a consequence of 2 and 3, because a family of sets that is closed under complements and disjoint countable unions does not need to be closed under countable unions: .)
If A is a Lebesgue-measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure.
A Lebesgue-measurable set can be "squeezed" between a containing open set and a contained closed set. This property has been used as an alternative definition of Lebesgue measurability. More precisely, is Lebesgue-measurable if and only if for every there exist an open set and a closed set such that and .[8]
A Lebesgue-measurable set can be "squeezed" between a containing
Gδ set and a contained
Fσ. I.e, if A is Lebesgue-measurable then there exist a
Gδ setG and an
FσF such that G ⊇ A ⊇ F and λ(G \ A) = λ(A \ F) = 0.
Lebesgue measure is
strictly positive on non-empty open sets, and so its
support is the whole of Rn.
If A is a Lebesgue-measurable set with λ(A) = 0 (a
null set), then every subset of A is also a null set.
A fortiori, every subset of A is measurable.
If A is Lebesgue-measurable and x is an element of Rn, 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.
If A is Lebesgue-measurable and , then the dilation of by defined by is also Lebesgue-measurable and has measure
More generally, if T is a
linear transformation and A is a measurable subset of Rn, then T(A) is also Lebesgue-measurable and has the measure .
All the above may be succinctly summarized as follows (although the last two assertions are non-trivially linked to the following):
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.
where bi ≥ ai, and the product symbol here represents a Cartesian product. The volume of this box is defined to be
For any subset A of Rn, we can define its
outer measureλ*(A) by:
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.
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.