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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. More often than not, not all subsets of a given set X are required to be associated to a given 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 notion of 'length', 'area' or 'volume'. The Lebesgue measure defines this for subsets of Euclidean space; but arbitrary measures generalize this notion for subsets of any set. The main use of the notion of a measure is to define the Lebesgue integral which has numerous applications in probability theory although there are other applications as well, particularly in mathematical analysis. There is a related notion of volume form used in differential topology.

Definition

Formally, a measure μ is a function (usually denoted by a Greek letter such as μ) defined on a σ-algebra Σ over a set X and taking values in the extended interval [0,∞] such that the following properties are satisfied:

The empty set has measure zero:

\mu (\varnothing )=0.

Countable additivity or σ-additivity: if E₁, E₂, E₃, … is a countable sequence 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 E_i:

\mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }\mu (E_{i}).

The triple (X, Σ, μ) is then called a measure space, and the members of Σ are called measurable sets.

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.

Properties

Several further properties can be derived from the definition of a countably additive measure.

Monotonicity

A measure μ is monotonic: If E₁ and E₂ are measurable sets with E₁ ⊆ E₂ then

\mu (E_{1})\leq \mu (E_{2}).

Measures of infinite unions of measurable sets

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

\mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)\leq \sum _{i=1}^{\infty }\mu (E_{i}).

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

\mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i}).

Measures of infinite intersections of measurable sets

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

\mu \left(\bigcap _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i}).

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

E_{n}=[n,\infty )\subseteq \mathbb {R}

which all have infinite measure, but the intersection is empty.

Sigma-finite measures

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.

Completeness

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).

Examples

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 μ_a (cf. Dirac delta function) is given by μ_a(S) = χ_S(a) = [a ∈ S], where χ_S is the characteristic function of S and the brackets signify the Iverson bracket. The measure of a set is 1 if it contains the point a and 0 otherwise.

Other measures include: Borel measure, Jordan measure, Ergodic measure, Euler measure, Gauss measure, Baire measure, Radon measure.

Non-measurable sets

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.

Generalizations

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.

References

R. G. Bartle, 1995. The Elements of Integration and Lebesgue Measure. Wiley Interscience.
Bourbaki, Nicolas (2004), Integration I, Springer Verlag, ISBN 3-540-41129-1 Chapter III.
R. M. Dudley, 2002. Real Analysis and Probability. Cambridge University Press.
Folland, Gerald B. (1999), Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0-471-317160-0 {{citation}}: Check |isbn= value: length (help) Second edition.
D. H. Fremlin, 2000. Measure Theory. Torres Fremlin.
Paul Halmos, 1950. Measure theory. Van Nostrand and Co.
R. Duncan Luce and Louis Narens (1987). "measurement, theory of," The New Palgrave: A Dictionary of Economics, v. 3, pp. 428-32.
M. E. Munroe, 1953. Introduction to Measure and Integration. Addison Wesley.
Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8. Emphasizes the Daniell integral.

External links

Measure theory for dummies, pdf article

@@ Line 1: / Line 1: @@
-[[Image:Measure illustration.png|right|thumb|Informally, a measure maps sets to non-negative real numbers, with supersets being mapped to bigger numbers.]]
+[[Image:Measure illustration.png|right|thumb|Informally, a measure has the property of being [[monotone function|monotone]] in the sense that if ''A'' is a [[subset]] of ''B'', the measure of ''A'' is less than or equal to the measure of ''B''. Furthermore, the measure of the [[empty set]] is required to be 0.]]
-In [[mathematics]] the concept of a '''measure''' generalizes notions such as "length", "area", and "volume" (but not all of its applications have to do with physical sizes). Informally, given some base [[Set (mathematics)|set]], a "measure" is any consistent assignment of "sizes" to (some of) the [[subset]]s of the base set. Depending on the application, the "size" of a subset may be interpreted as (for example) its physical size, the amount of something that lies within the subset, or the probability that some random process will yield a result within the subset. The main use of measures is to define general concepts of [[integral|integration]] over domains with more complex structure than intervals of the real line. Such integrals are used extensively in [[probability theory]], and in much of [[mathematical analysis]].
+In [[mathematics]], more specifically [[measure theory]], a '''measure''' is intuitively a certain association between [[subset]]s of a given [[set]] ''X'' and the ([[extended real number system|extended set]]) of non-negative real numbers. More often than not, not all subsets of a given set ''X'' are required to be associated to a given non-negative real number; the subsets which are required to be associated to a non-negative real number are known as the [[measurable set|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.
-It is often not possible or desirable to assign a size to ''all'' subsets of the base set, so a measure does not have to do so. There are certain consistency conditions that govern which combinations of subsets it is allowed for a measure to assign sizes to; these conditions are encapsulated in the auxiliary concept of a [[sigma-algebra|σ-algebra]].
+Measures can be thought of as a generalization of the notion of 'length', 'area' or 'volume'. The [[Lebesgue measure]] defines this for subsets of [[Euclidean space]]; but arbitrary measures generalize this notion for subsets of any set. The main use of the notion of a measure is to define the [[Lebesgue integral]] which has numerous applications in [[probability theory]] although there are other applications as well, particularly in [[mathematical analysis]]. There is a related notion of [[volume form]] used in [[differential topology]].
-In [[differential topology]], the related concept of [[volume form]] is used more frequently.
-'''Measure theory''' is a branch of [[real analysis]] which investigates [[sigma-algebra|σ-algebras]], measures, [[measurable function]]s and [[integral]]s.
 ==Definition==
-Formally, a measure &mu; is a [[function (mathematics)|function]] defined on a [[sigma-algebra|σ-algebra]] &Sigma; over a set ''X'' and taking values in the [[extended real line|extended interval]] [0,&infin;] such that the following properties are satisfied:
+Formally, a measure &mu; is a [[function (mathematics)|function]] (usually denoted by a Greek letter such as μ) defined on a [[sigma-algebra|σ-algebra]] &Sigma; over a set ''X'' and taking values in the [[extended real line|extended interval]] [0,&infin;] such that the following properties are satisfied:
 * The [[empty set]] has [[measure zero]]: