Lebesgue measure

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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.[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 denoted by λ(A).

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 dx, but this should not be confused with the distinct notion of a volume form.

Definition[edit]

Given a subset E\subset\mathbb{R}, with the length of an (open, closed, semi-open) interval I = [a,b] given by l(I)=b - a, the Lebesgue outer measure \lambda^*(E) is defined as

\lambda^*(E) = \text{inf} \left\{\sum_{k=1}^\infty l(I_k) : {(I_k)_{k \in \mathbb N}} \text{ is a sequence of open intervals with } E\subset \bigcup_{k=1}^\infty I_k\right\}.

The Lebesgue measure of E is given by its Lebesgue outer measure \lambda(E)=\lambda^*(E) if, for every  A\subset\mathbb{R},

\lambda^*(A) = \lambda^*(A \cap E) + \lambda^*(A \cap E^c) .

Intuition[edit]

The first part of the definition states that the subset E of the real numbers is reduced to its outer measure by coverage by sets of intervals. Each of these sets of intervals I covers E in the sense that when the intervals are combined together by union, they form a superset of E. Moreover, the intervals in each set are disjoint, and there is a countable infinity of these intervals. For each set, the total length is calculated by adding the lengths of this infinity of disjoint intervals. This total length of any interval set can easily overestimate the measure of E, because E is a subset of the union of the intervals, and so the intervals may include points which are not in E. 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 E most tightly.

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 A using E as an instrument to split A into two partitions: the part of A which intersects with E and the remaining part of A which is not in E: the set difference of A and E. These partitions of A are subject to the outer measure. If for all possible such subsets A of the real numbers, the partitions of A cut apart by E have outer measures which add up to the outer measure of A, then the outer Lebesgue measure of E gives its Lebesgue measure. Intuitively, this condition means that the set E must not have some curious properties which causes a discrepancy in the measure of another set when E 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[edit]

Properties[edit]

Translation invariance: The Lebesgue measure of A and A+t 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 \lambda (A)=|I_1|\cdot |I_2|\cdots |I_n|. 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 need not be closed under countable unions: \{\emptyset, \{1,2,3,4\}, \{1,2\}, \{3,4\}, \{1,3\}, \{2,4\}\}.)
  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. Lebesgue measure is both locally finite and inner regular, and so it is a Radon measure.
  10. Lebesgue measure is strictly positive on non-empty open sets, and so its support is the whole of Rn.
  11. 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.
  12. 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.
  13. If A is Lebesgue measurable and \delta>0, then the dilation of A by \delta defined by \delta A=\{\delta x:x\in A\} is also Lebesgue measurable and has measure \delta^{n}\lambda\,(A).
  14. 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)|\, \lambda\,(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 with \lambda([0,1]\times [0, 1]\times \cdots \times [0, 1])=1.

The Lebesgue measure also has the property of being σ-finite.

Null sets[edit]

Main article: Null set

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[disambiguation needed] 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) \cup(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[edit]

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

B=\prod_{i=1}^n [a_i,b_i] \, ,

where biai, and the product symbol here represents a Cartesian product. The volume of this box is defined to be

\operatorname{vol}(B)=\prod_{i=1}^n (b_i-a_i) \, .

For any subset A of Rn, we can define its outer measure λ*(A) by:

\lambda^*(A) = \inf \Bigl\{\sum_{B\in \mathcal{C}}\operatorname{vol}(B) : \mathcal{C}\text{ is a countable collection of boxes whose union covers }A\Bigr\} .

We then define the set A to be Lebesgue measurable if for every subset S of Rn,

\lambda^*(S) = \lambda^*(S \cap A) + \lambda^*(S \setminus A) \, .

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).[3]

Relation to other measures[edit]

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[edit]

References[edit]

  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. 
  3. ^ 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.