In mathematics, more specifically measure theory, a measure is intuitively a certain association between subsets of a given set X and the (extended set) of non-negative real numbers. Often, some subsets of a given set X are not required to be associated to a non-negative real number; the subsets which are required to be associated to a non-negative real number are known as the measurable subsets of X. The collection of all measurable subsets of X is required to form what is known as a sigma algebra; namely, a sigma algebra is a subcollection of the collection of all subsets of X that in addition, satisfies certain axioms.

Measures can be thought of as a generalization of the notions: 'length,' 'area' and 'volume.' The Lebesgue measure defines this for subsets of a Euclidean space, and an arbitrary measure generalizes this notion to subsets of any set. The original intent for measure was to define the Lebesgue integral, which increases the set of integrable functions considerably. It has since found numerous applications in probability theory, in addition to several other areas of academia, particularly in mathematical analysis. There is a related notion of volume form used in differential topology.

Suppose that is a set and is a σ-algebra over . Then a measure μ is a function with domain Σ and codomain (see extended interval) such that the following property are satisfied:

• Positivity, i.e. μ() ≥ 0 for all in Σ;

• Null emptyset, i.e. μ(0)=0 (where 0 denotes the empty set);

• Countable additivity or σ-additivity: if is a countable set of pairwise disjoint sets in Σ, the measure of the union of all the E_i is equal to the sum of the measures of each :

Furthermore we call the triple (X, Σ, μ) a measure space, and the members of Σ are called measurable sets.

If 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 E₁ and E₂ are measurable sets with E₁ ⊆ E₂ then

A measure μ is countably subadditive: If E₁, E₂, E₃, … is a countable sequence of sets in Σ, not necessarily disjoint, then

A measure μ is continuous from below: If E₁, E₂, E₃, … 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 E₁, E₂, E₃, … 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 measure, but the intersection is empty.

Main article: Sigma-finite measure

A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). It is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. A set in a measure space has σ-finite measure if it is a countable union of sets with finite measure.

For example, the real numbers with the standard Lebesgue measure are σ-finite 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 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 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 σ-algebra 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.

• The counting measure is defined by μ(S) = number of elements in S.
• The Lebesgue measure on R is a complete translation-invariant measure on a σ-algebra containing the intervals in R such that μ([0,1]) = 1; and every other measure with these properties extends Lebesgue measure.
• Circular angle measure is invariant under rotation.
• The Haar measure for a 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 which is a refinement of the Lebesgue measure to some 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 characteristic function of S. The measure of a set is 1 if it contains the point a and 0 otherwise.

Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gauss measure, Baire measure, Radon measure.

Main article: Non-measurable set

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 non-measurable 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 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 mainly 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.

Another generalization is the finitely additive measure. 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.

The remarkable result in integral geometry known as Hadwiger's theorem states that the space of translation-invariant, finitely additive, not-necessarily-nonnegative set functions defined on finite unions of compact convex sets in Rⁿ 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 n-dimensional 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.

A measure is a special kind of content.

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• Measure theory for dummies, pdf article