Lebesgue measure

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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 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. [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).

Contents

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. [2]

The Lebesgue measure is often denoted by dx, but this should not be confused with the distinct notion of a volume form.

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

Some sets satisfy the Carathéodory criterion, which requires that for every ,

The set of all such forms a σ-algebra. For any such , its Lebesgue measure is defined to be its Lebesgue outer measure: .

A set that does not satisfy the Carathéodory criterion is not Lebesgue-measurable. Non-measurable sets do exist; an example is the Vitali sets.

Intuition

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

Properties

Translation invariance: The Lebesgue measure of
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Translation invariance: The Lebesgue measure of and are the same.

The Lebesgue measure on Rn has the following properties:

  1. If A is a cartesian product of intervals I1 × I2 × ⋯ × In, then A is Lebesgue-measurable and Here, |I| denotes the length of the interval I.
  2. 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.
  3. If A is Lebesgue-measurable, then so is its complement.
  4. λ(A) ≥ 0 for every Lebesgue-measurable set A.
  5. 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.)
  6. 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: .)
  7. If A is an open or closed subset of Rn (or even Borel set, see metric space), then A is Lebesgue-measurable.
  8. 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).
  9. 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 . [7]
  10. 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 set G and an Fσ F such that G  A  F and λ(G \ A) = λ(A \ F) = 0.
  11. Lebesgue measure is both locally finite and inner regular, and so it is a Radon measure.
  12. Lebesgue measure is strictly positive on non-empty open sets, and so its support is the whole of Rn.
  13. 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.
  14. 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 : aA}, is also Lebesgue-measurable and has the same measure as A.
  15. If A is Lebesgue-measurable and , then the dilation of by defined by is also Lebesgue-measurable and has measure
  16. 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):

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

The 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 (AB) ∪ (BA) is a null set) and then show 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 is an application of Carathéodory's extension theorem. It proceeds as follows.

Fix nN. A box in Rn is a set of the form

where biai, 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 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). [8]

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.

See also

Related Research Articles

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References

  1. The term volume is also used, more strictly, as a synonym of 3-dimensional volume
  2. Henri Lebesgue (1902). "Intégrale, longueur, aire". Université de Paris.Cite journal requires |journal= (help)
  3. Royden, H. L. (1988). Real Analysis (3rd ed.). New York: Macmillan. p. 56. ISBN   0-02-404151-3.
  4. Asaf Karagila. "What sets are Lebesgue-measurable?". math stack exchange. Retrieved 26 September 2015.
  5. Asaf Karagila. "Is there a sigma-algebra on R strictly between the Borel and Lebesgue algebras?". math stack exchange. Retrieved 26 September 2015.
  6. Osgood, William F. (January 1903). "A Jordan Curve of Positive Area". Transactions of the American Mathematical Society. American Mathematical Society. 4 (1): 107–112. doi: 10.2307/1986455 . ISSN   0002-9947. JSTOR   1986455.
  7. Carothers, N. L. (2000). Real Analysis. Cambridge: Cambridge University Press. pp.  293. ISBN   9780521497565.
  8. Solovay, Robert M. (1970). "A model of set-theory in which every set of reals is Lebesgue-measurable". Annals of Mathematics . Second Series. 92 (1): 1–56. doi:10.2307/1970696. JSTOR   1970696.