DefinitionLet be a set and a -algebra over . A function from to the extended real number line is called a measure if it satisfies the following properties: *Non-negativity: For all in Σ, we have . *Null empty set: . *Countable additivity (or sigma additivity, -additivity): For all countable collections of pairwise disjoint sets in Σ, : If at least one set has finite measure, then the requirement that is met automatically. Indeed, by countable additivity, : and therefore If only the second and third conditions of the definition of measure above are met, and takes on at most one of the values , then is called a signed measure. The pair is called a measurable space, the members of Σ are called measurable sets. If and are two measurable spaces, then a function is called measurable if for every -measurable set , the Image (mathematics)#Inverse image, inverse image is -measurable – i.e.: . In this setup, the Function composition, composition of measurable functions is measurable, making the measurable spaces and measurable functions a Category (mathematics), category, with the measurable spaces as objects and the set of measurable functions as arrows. See also Measurable function#Term usage variations about another setup. A tuple, triple is called a measure space. A probability measure is a measure with total measure one – i.e. . 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 (mathematics), 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 support (mathematics)#Compact support, compact support. This approach is taken by Nicolas Bourbaki, Bourbaki (2004) and a number of other sources. For more details, see the article on Radon measures.
InstancesSome important measures are listed here. * The counting measure is defined by = number of elements in . * The Lebesgue measure on is a Complete measure, complete translational invariance, translation-invariant measure on a ''σ''-algebra containing the interval (mathematics), intervals in such that ; and every other measure with these properties extends Lebesgue measure. * Circular angle measure is invariant under rotation, and hyperbolic angle measure is invariant under squeeze mapping. * The Haar measure for a Locally compact space, locally compact topological group is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties. * The Hausdorff measure is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets. * Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval [0, 1]). Such a measure is called a ''probability measure''. See probability axioms. * The Dirac measure δ''a'' (cf. Dirac delta function) is given by δ''a''(''S'') = χ''S''(a), where χ''S'' is the indicator function of . The measure of a set is 1 if it contains the point and 0 otherwise. Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure, Young measure, and Loeb measure. In physics an example of a measure is spatial distribution of mass (see e.g., gravity potential), or another non-negative extensive property, conserved quantity, conserved (see Conservation law (physics), conservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below. * Liouville's theorem (Hamiltonian)#Symplectic geometry, 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.
Basic propertiesLet be a measure.
MonotonicityIf and are measurable sets with then :
Measure of countable unions and intersections
SubadditivityFor any countable Sequence (mathematics), sequence of (not necessarily disjoint) measurable sets in Σ: :
Continuity from belowIf are measurable sets and for all , then the Union (set theory), union of the sets is measurable, and :
Continuity from aboveIf are measurable sets and, for all , then the Intersection (set theory), intersection of the sets is measurable; furthermore, if at least one of the has finite measure, then : This property is false without the assumption that at least one of the has finite measure. For instance, for each , let , which all have infinite Lebesgue measure, but the intersection is empty.
CompletenessA measurable set is called a ''null set'' if . 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 σ-algebra of subsets which differ by a negligible set from a measurable set , that is, such that the symmetric difference of and is contained in a null set. One defines to equal .
AdditivityMeasures are required to be countably additive. However, the condition can be strengthened as follows. For any set and any set of nonnegative define: : That is, we define the sum of the to be the supremum of all the sums of finitely many of them. A measure on is -additive if for any and any family of disjoint sets the following hold: : : Note that the second condition is equivalent to the statement that the Ideal (set theory), ideal of null sets is -complete.
Sigma-finite measuresA measure space is called finite if is a finite real number (rather than ∞). Nonzero finite measures are analogous to probability measures in the sense that any finite measure is proportional to the probability measure . A measure is called ''σ-finite'' if can be decomposed into a countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a ''σ-finite measure'' if it is a countable union of sets with finite measure. For example, the s with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals for all integers ; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the s with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, 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 σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf space, 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'.
s-finite measuresA measure is said to be s-finite if it is a countable sum of bounded measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes.
Non-measurable setsIf the axiom of choice is assumed to be true, it can be proved that not all subsets of are Lebesgue measurable; examples of such sets include the Vitali set, and the non-measurable sets postulated by the Hausdorff paradox and the Banach–Tarski paradox.
GeneralizationsFor certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative 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 self-adjoint projections on a Hilbert space is called a ''projection-valued measure''; these are used in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative 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'', also known as a Content (measure theory), content. 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. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of lp space, ''L''∞ and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice. Contents remain useful in certain technical problems in geometric measure theory; this is the theory of Banach measures. A signed measure, charge is a generalization in both directions: it is a finitely additive, signed measure.
See also* Abelian von Neumann algebra * Almost everywhere * Carathéodory's extension theorem * Content (measure theory) * Fubini's theorem * Fatou's lemma * Fuzzy measure theory * Geometric measure theory * Hausdorff measure * Inner measure * Lebesgue integration * Lebesgue measure * Lorentz space * Lifting theory * Measurable cardinal * Measurable function * Minkowski content * Outer measure * Product measure * Pushforward measure * Regular measure * Vector measure * Valuation (measure theory) * Volume form
Bibliography* Robert G. Bartle (1995) ''The Elements of Integration and Lebesgue Measure'', Wiley Interscience. * * * * Chapter III. * R. M. Dudley, 2002. ''Real Analysis and Probability''. Cambridge University Press. * Second edition. * Federer, Herbert. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969 xiv+676 pp. * D. H. Fremlin, 2000.