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Measure (mathematics)
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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. Measures can thus be thought of as a generalization of the notions of 'length,' 'area' and 'volume.' There are in general infinitely many different measures on a given set, each assigning different "sizes" for subsets.
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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. Measures can thus be thought of as a generalization of the notions of 'length,' 'area' and 'volume.' There are in general infinitely many different measures on a given set, each assigning different "sizes" for subsets. A particularly important example is the Lebesgue measure on an Euclidean space, assigning the 'usual' length, area and volume to subsets. For instance, the Lebesgue measure of the interval [0,1] of the real line is its length, i.e., one.
A function assigning a positive real number or infinity to subsets of a set has to satisfy a few conditions to qualify as a measure; in particular, the 'size' of the union of a sequence of subsets that are disjoint from each other has to be 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, also satisfying the condition above. This problem is resolved by defining the measure only for 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 sigma algebra, meaning that unions, intersections and complements of sequences of measurable subsets are again measurable. Non-measurable sets in an Euclidean space, on which the Lebesgue measure cannot be consistently defined, are necessarily extremely complex, in a sense badly mixed up with their complement; indeed, their existence is a non-trivial consequence of the axiom of choice.
The main applications of the measures, developed in successive stages in late 19th and early 20th century by, among others, Emile Borel, Henri Lebesgue, Johann Radon and Maurice Fréchet, are found in the foundations of Lebesgue integral and (in a closely related manner) Kolmogorov's axiomatisation of probability theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of the Euclidean space; moreover, integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than the preceding theory of Riemann integral. Probability theory considers measures that assign to the whole set the 'size' one, and considers measurable subsets to be 'events' with probability given by the measure.
Definition
Suppose that is a set and is a s-algebra over . Then a measure µ is a function with domain Σ and codomain (see extended interval) such that the following properties are satisfied:
- Non-negativity, i.e. µ = 0 for all in S;
- Null empty set, i.e. µ(Ř) = 0 (where Ř denotes the empty set);
- Countable additivity or s-additivity: if is a countable set of pairwise disjoint sets in Σ, the measure of the union of all the Ei 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. The second condition actually follows from the third condition assuming that there is at least one set having finite 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.
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
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
A measure μ is continuous from below: If E1, E2, E3, … are measurable sets and En is a subset of En + 1 for all n, then the union of the sets En is measurable, and
Measures of infinite intersections of measurable sets
A measure μ is continuous from above: If E1, E2, E3, … are measurable sets and En + 1 is a subset of En for all n, then the intersection of the sets En is measurable; furthermore, if at least one of the En has finite measure, then
This property is false without the assumption that at least one of the En has finite measure. For instance, for each n ∈ N, let
which all have infinite measure, but the intersection is empty.
Sigma-finite measures
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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'.
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 s-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 δa (cf. Dirac delta function) is given by δa(S) = χS(a), where χS 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.
Non-measurable sets
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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.
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. 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. 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 L8 and the Stone–Cech 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 ck. 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.
See also
External links
- [https://www.ee.washington.edu/techsite/papers/documents/UWEETR-2006-0008.pdf Measure theory for dummies, pdf article]
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