![]() | ||
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For 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, or simply volume. 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 denoted by λ(A).
Contents
- Definition
- Intuition
- Examples
- Properties
- Null sets
- Construction of the Lebesgue measure
- Relation to other measures
- References
Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.
The Lebesgue measure is often denoted dx, but this should not be confused with the distinct notion of a volume form.
Definition
Given a subset
The Lebesgue measure is defined on the Lebesgue σ-algebra, which is the collection of all the sets E which satisfy the condition that, for every
For any set in the Lebesgue σ-algebra, its Lebesgue measure is given by its Lebesgue outer measure
Sets that aren't included in the Lebesgue σ-algebra are not Lebesgue measurable. Such sets do exist, i.e., Lebesgue σ-algebra is strictly contained in the power set of
Intuition
The first part of the definition states that the subset
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
Examples
Properties
The Lebesgue measure on Rn has the following properties:
- If A is a cartesian product of intervals I1 × I2 × ... × In, then A is Lebesgue measurable and
λ ( A ) = | I 1 | ⋅ | I 2 | ⋯ | I n | . Here, |I| denotes the length of the interval I. - 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, 3 and 4.)
- 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 need not be closed under countable unions:
{ ∅ , { 1 , 2 , 3 , 4 } , { 1 , 2 } , { 3 , 4 } , { 1 , 3 } , { 2 , 4 } } .) - If A is an open or closed subset of Rn (or even Borel set, see metric space), then A is Lebesgue measurable.
- If A is a Lebesgue measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure (see the regularity theorem for 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,
E ⊂ R is Lebesgue measurable if and only if for everyϵ > 0 there exist an open setG and a closed setF such thatF ⊂ E ⊂ G andλ ( G ∖ F ) < ϵ . - A Lebesgue-measurable set can be "squeezed" between a containing Gδ set and a contained Fσ set. I.e, if A is Lebesgue measurable then there exist a Gδ set G and an Fσ set F such that G⊇A⊇F and λ(GA)=λ(AF)=0.
- Lebesgue measure is both locally finite and inner regular, and so it is a Radon measure.
- 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
δ > 0 , then the dilation ofA byδ defined byδ A = { δ x : x ∈ A } is also Lebesgue measurable and has measureδ n λ ( A ) . - 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
| det ( T ) | λ ( 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 withThe Lebesgue measure also has the property of being σ-finite.
Null sets
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)
Construction of the Lebesgue measure
The modern construction of the Lebesgue measure is an application of Carathéodory's extension theorem. It proceeds as follows.
Fix n ∈ N. A box in Rn is a set of the form
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 measure λ*(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 a certain set-theoretical axiom, the 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).
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 (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.
It can be shown that there is no infinite-dimensional analogue of Lebesgue measure.