Lebesgue measure on arvau

Wikipedia / Lebesgue measure

In mathematics, Lebesgue measure is the standard way of assigning a notion of length to subsets of the real line, area to regions of the Euclidean plane, and volume to subsets of Euclidean space, it is used throughout mathematical analysis, especially in the definition of the Lebesgue integral and in statements that hold "almost everywhere," meaning except on a set whose Lebesgue measure is zero.

Lebesgue measure extends ordinary geometric length (or volume) in a way that is compatible with countable unions and other kinds of countable limits of sets. For example, every countable subset of the real line has Lebesgue measure zero, being a countable union of points, which have no length, while many uncountable sets also have measure zero. The measure is not defined on every subset of the real line (or Euclidean space) under the usual axioms of set theory: the sets to which it applies are called Lebesgue-measurable.

One way to characterize the Lebesgue measure is to first define it on Borel sets, that is all sets that can be obtained by countably many operations of unions, set complement, and intersections, from the collection open intervals, so that it assigns the usual length to open intervals, and satisfies natural properties under taking limits of intervals. The Lebesgue measure can then be obtained by completing this Borel measure, by assigning zero measure to all subsets of Borel sets that already have zero measure.

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 Intégrale, Longueur, Aire in 1902.

Elementary construction

A construction related to the Lebesgue measure is the Jordan content, which approximates regions by finite partitions into rectangular boxes. Similar to the Riemann integral, a set is Jordan measurable if there are such partitions that contain the region, and partitions contained within the region, that have an arbitrarily small difference between them. Jordan measure is not as robust as the Lebesgue measure, because there are fairly basic sets that are not Jordan measurable, such as the set of rational numbers.

Lebesgue measure extends this assignment from intervals and boxes to a sufficiently large class of more complicated sets while preserving the basic rule of countable additivity This requirement is stronger than finite additivity can be assigned a Lebesgue measure in the usual set-theoretic setting.

The first domain for the measure is the collection of Borel sets form the smallest \sigma-algebra containing all open sets. Equivalently, they are the sets that can be obtained from open sets by applying countable unions, countable intersections, and complements. This includes open sets, closed sets, countable sets, intervals, boxes, and many other sets obtained from them by countable operations (e.g., the Cantor ternary set).

that assigns to every rectangular box its usual volume and is invariant under translations. This is often called the Borel version of Lebesgue measure. However, as a measure on only the Borel sets, it is not complete: a subset of a Borel set of measure zero need not itself be Borel.

. This is well-defined because changing a set by a null set does not change its measure.

-algebra than the Borel sets.

Carathéodory characterization

For any interval, the Lebesgue outer measure is defined as an infimum

The above definition can be generalised to higher dimensions as follows. For any rectangular cuboid which is a Cartesian product,

satisfies the Carathéodory criterion, we have:

satisfying the Carathéodory criterion are said to be Lebesgue-measurable forms a σ-algebra.

The Lebesgue measure of such a set is defined as its Lebesgue outer measure: . ZFC proves that non-measurable sets do exist; examples are the Vitali sets.

Intuition

. The Lebesgue outer measure emerges as the greatest lower bound (infimum) most tightly and do not overlap.

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 interval of real numbers. The open interval has the same measure, since the difference, which each have measure zero.
Any Cartesian product, the area of the corresponding rectangle.
Moreover, every Borel set is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borel sets.
Any countable set of real numbers has Lebesgue measure . In particular, the Lebesgue measure of the set of algebraic numbers is , even though the set is dense.
The Cantor set and the set of Liouville numbers are examples of uncountable sets that have Lebesgue measure .
If the axiom of determinacy holds then all sets of reals are Lebesgue-measurable. Determinacy is however not compatible with the axiom of choice.
Vitali sets are examples of sets that are not measurable with respect to the Lebesgue measure. Their existence relies on the axiom of choice.
Osgood curves are simple plane curves with positive Lebesgue measure (it can be obtained by small variation of the Peano curve construction). The dragon curve is another unusual example.
, has a zero Lebesgue measure. In general, every proper hyperplane has a zero Lebesgue measure in its ambient space.
The volume of an n-ball can be calculated in terms of Euler's gamma function.

Properties

has the following properties:

is a cartesian product of intervals, then A
If is a union of countably many pairwise disjoint Lebesgue-measurable sets, then is itself Lebesgue-measurable and is equal to the sum (or infinite series) of the measures of the involved measurable sets.
If is Lebesgue-measurable, then so is its complement.
for every Lebesgue-measurable set .
If and are Lebesgue-measurable and is a subset of , then . (A consequence of 2.)
Countable unions and intersections.)
If is an open or closed (or even Borel set, see metric space), then is Lebesgue-measurable.
If 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 G δ set and a contained F σ. I.e., if is Lebesgue-measurable then there exist a G δ set and an F σ such that and .
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.
If is a Lebesgue-measurable set with (a null set), then every subset of is also a null set. A fortiori is measurable.
If is Lebesgue-measurable and x, then the , is also Lebesgue-measurable and has the same measure as .
If , then the
More generally, if is a linear transformation and , then .

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.

Null sets

is a null set, it can be covered with countably many products of n. All countable sets are null sets.

has Hausdorff dimension less than then it is a null set with respect to -dimensional Lebesgue measure. Here Hausdorff dimension is relative to the Euclidean metric (or any metric Lipschitz equivalent to it). On the other hand, a set may have topological dimension less than and have positive -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 is Lebesgue-measurable, one usually tries to find a "nicer" set which differs from only by a null set (in the sense that the symmetric difference is a null set) and then show that 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.

. A boxFor any subset , we can define its outer measureWe then define the set to be Lebesgue-measurable if for every subset These Lebesgue-measurable sets form a σ-algebra for any Lebesgue-measurable set .

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 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 is automatically a locally finite is necessarily a Lebesgue measure. The Borel measure is translation-invariant, but not complete.

The Haar measure can be defined on any locally compact group with addition is a locally compact group).

The Hausdorff measure of lower dimensions than , like submanifolds 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.