In mathematics, a measure is a function that assigns "sizes", "volumes", or "probabilities" to subsets of a given set. The concept is important in mathematical analysis and probability theory.
Measure theory is that branch of real analysis which investigates sigma algebras, measures, measurable functions and integrals. It is of importance in probability and statistics.
See also Lebesgue integration,Lebesgue measure
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Formal definitions
Formally, a measure μ is a function which assigns to every element S of a given sigma algebra X a value μ(S), a nonnegative real number or ∞. The following properties have to be satisfied:
 The empty set has measure zero: μ({}) = 0.
 The measure is countably additive: if E_{1}, E_{2}, E_{3}, ... are countably many pairwise disjoint sets in X and E is their union, then the measure μ(E) is equal to the sum ∑μ(E_{k}).
The following properties can be derived from the definition above:
 If E_{1} and E_{2} are two measurable sets with E_{1} being a subset of E_{2}, then μ(E_{1}) ≤ μ(E_{2}).
 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 E of the sets E_{n} is measurable and μ(E) = lim μ(E_{n}).
 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 E of the sets E_{n} is measurable; furthermore, if at least one of the E_{n} has finite measure, then μ(E) = lim μ(E_{n}).
σfinite measure spaces have some very nice properties; σfiniteness can be compared in this respect to separable of topological spaces.
A measurable set S is called a nullset if μ(S) = 0. The measure μ is called complete if every subset of a nullset is measurable (and then automatically itself a nullset).
Examples
Some important measures are listed here.
 The counting measure is defined by μ(S) = number of elements in S.
 The Lebesgue measure is the unique complete translationinvariant measure on a sigma algebra containing the intervalss in R such that μ([0,1]) = 1.
 The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure and has a similar uniqueness property.
 The zero measure is defined by μ(S) = 0 for all S.
 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.
Generalizations
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. A measure that takes values in a Banach space is called a spectral measure; these are used mainly in functional analysis for the spectral theorem.
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.
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 n1 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 homogenous of degree 0 is the Euler characteristic.