In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, volume, et cetera. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area and volume of Euclidean geometry to suitable subsets of R^{n}, n=1,2,3,.... For instance, the Lebesgue measure of [0,1] in the real numbers is its length in the everyday sense of the word, specifically 1.
To qualify as a measure (see Definition below), a function that assigns a nonnegative real number or +∞ to a set's subsets must satisfy a few conditions. One important condition is countable additivity. This condition states that the size of the union of a sequence of disjoint subsets is equal to the sum of the sizes of the subsets. However, it is in general impossible to consistently associate a size to each subset of a given set and also satisfy the other axioms of a measure. This problem was resolved by defining measure only on a subcollection of all subsets; the subsets on which the measure is to be defined are called measurable and they are required to form a sigmaalgebra, meaning that unions, intersections and complements of sequences of measurable subsets are measurable. Nonmeasurable sets in a Euclidean space, on which the Lebesgue measure cannot be consistently defined, are necessarily complex to the point of incomprehensibility, in a sense badly mixed up with their complement; indeed, their existence is a nontrivial consequence of the axiom of choice.
Measure theory was developed in successive stages during the late 19th and early 20th century by Emile Borel, Henri Lebesgue, Johann Radon and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.
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Let Σ be a σalgebra over a set X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties:
The second condition may be treated as a special case of countable additivity, if the empty collection is allowed as a countable collection (and the empty sum is interpreted as 0). Otherwise, if the empty collection is disallowed (but finite collections are allowed), the second condition still follows from countable additivity provided, however, that there is at least one set having finite measure.
The pair (X, Σ) is called a measurable space, the members of Σ are called measurable sets, and the triple (X, Σ, μ) is called a measure space.
If only the second and third conditions are met, and μ takes on at most one of the values ±∞, then μ is called a signed measure.
A probability measure is a measure with total measure one (i.e., μ(X) = 1); a probability space is a measure space with a probability measure.
For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other authors. For more details see Radon measure.
Several further properties can be derived from the definition of a countably additive measure.
A measure μ is monotonic: If E_{1} and E_{2} are measurable sets with E_{1} ⊆ E_{2} then
A measure μ is countably subadditive: If E_{1}, E_{2}, E_{3}, … is a countable sequence of sets in Σ, not necessarily disjoint, then
A measure μ is continuous from below: If E_{1}, E_{2}, E_{3}, … are measurable sets and E_{n} is a subset of E_{n + 1} for all n, then the union of the sets E_{n} is measurable, and
A measure μ is continuous from above: If E_{1}, E_{2}, E_{3}, … are measurable sets and E_{n + 1} is a subset of E_{n} for all n, then the intersection of the sets E_{n} is measurable; furthermore, if at least one of the E_{n} has finite measure, then
This property is false without the assumption that at least one of the E_{n} has finite measure. For instance, for each n ∈ N, let
which all have infinite Lebesgue measure, but the intersection is empty.
A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). It is called σfinite if X can be decomposed into a countable union of measurable sets of finite measure. A set in a measure space has σfinite measure if it is a countable union of sets with finite measure.
For example, the real numbers with the standard Lebesgue measure are σfinite but not finite. Consider the closed intervals [k,k+1] for all integers k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σfinite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σfinite measure spaces have some very convenient properties; σfiniteness can be compared in this respect to the Lindelöf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.
A measurable set X is called a null set if μ(X)=0. A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable.
A measure can be extended to a complete one by considering the σalgebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. One defines μ(Y) to equal μ(X).
Some important measures are listed here.
Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure.
In physics an example of a measure is spatial distribution of mass (see e.g gravity potential), or another nonnegative extensive property, conserved (see conservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below.
Liouville measure, known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics.
Gibbs measure is widely used in statistical mechanics, often under the name canonical ensemble.
If the axiom of choice is assumed to be true, not all subsets of Euclidean space are Lebesgue measurable; examples of such sets include the Vitali set, and the nonmeasurable sets postulated by the Hausdorff paradox and the Banach–Tarski paradox.
For certain purposes, it is useful to have a "measure" whose values are not restricted to the nonnegative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. Measures that take values in Banach spaces have been studied extensively. A measure that takes values in the set of selfadjoint projections on a Hilbert space is called a projectionvalued measure; these are used mainly in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take nonnegative values from generalizations, the term positive measure is used. Positive measures are closed under conical combination but not general linear combination, while signed measures are the linear closure of positive measures.
Another generalization is the finitely additive measure, which are sometimes called contents. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first, but proved to be not so useful. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of L^{∞} and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice.
A charge is a generalization in both directions: it is a finitely additive, signed measure.
The remarkable result in integral geometry known as Hadwiger's theorem states that the space of translationinvariant, finitely additive, notnecessarilynonnegative set functions defined on finite unions of compact convex sets in R^{n} consists (up to scalar multiples) of one "measure" that is "homogeneous of degree k" for each k = 0, 1, 2, ..., n, and linear combinations of those "measures". "Homogeneous of degree k" means that rescaling any set by any factor c > 0 multiplies the set's "measure" by c^{k}. The one that is homogeneous of degree n is the ordinary ndimensional volume. The one that is homogeneous of degree n − 1 is the "surface volume". The one that is homogeneous of degree 1 is a mysterious function called the "mean width", a misnomer. The one that is homogeneous of degree 0 is the Euler characteristic.
