Information about Measure (mathematics)
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, a "measure" is any consistent assignment of "sizes" to (some of) the subsets 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 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.
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 σ-algebra.
Measure theory is that branch of real analysis which investigates σ-algebras, measures, measurable functions and integrals.
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.
is monotonic: If 
and 
are measurable sets with 
then 
.
is subadditive: If , , 
, ... is a countable sequence of sets in 
, not necessarily disjoint, then
is continuous from below: If , , , ... are measurable sets and 
is a subset of 
for all n, then the union of the sets is measurable, and
is continuous from above: If , , , ... are measurable sets and is a subset of for all n, then the 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 n ∈ N, let
which all have infinite measure, but the intersection is empty.
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 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 separability of topological spaces.
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).
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.
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
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 
multiplies the set's "measure" by 
. 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.
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 σ-algebra.
Measure theory is that branch of real analysis which investigates σ-algebras, measures, measurable functions and integrals.
)
Informally, a measure maps sets to non-negative real numbers, with larger sets being mapped to bigger numbers.
Definition
Formally, a measure μ is a function 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:
.
- Countable additivity or σ-additivity: if
... is a countable sequence of pairwise disjoint sets in , the measure of the union of all the 
's is equal to the sum of the measures of each :
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
is monotonic: If and are measurable sets with then .
Measures of infinite unions of measurable sets
is subadditive: If , , , ... is a countable sequence of sets in , not necessarily disjoint, then
.
is continuous from below: If , , , ... are measurable sets and is a subset of for all n, then the union of the sets is measurable, and
.
Measures of infinite intersections of measurable sets
is continuous from above: If , , , ... are measurable sets and is a subset of for all n, then the 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 n ∈ N, let
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 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 separability 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 is the unique complete translation-invariant measure on a σ-algebra containing the intervals in R such that μ([0,1]) = 1.
- Circular angle measure is invariant under rotation.
- The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure and has a similar uniqueness property.
- 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
(confer Dirac delta function) is given by 
, where 
is the characteristic function of S. The measure of a set is 1 if it contains the point a and 0 otherwise.
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
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 multiplies the set's "measure" by . 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.
See also
References
- R. G. Bartle, 1995. The Elements of Integration and Lebesgue Measure. Wiley Interscience.
-
id="CITEREFBourbaki2004">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.
-
id="CITEREFFolland1999">Folland, Gerald B. (1999), Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0-471-317160-0 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 , 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.
- Some useful Cambridge Tripos Notes on Probability and Measure Theory link
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
..... Click the link for more information.SET may stand for:- Sanlih Entertainment Television, a television channel in Taiwan
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..... Click the link for more information.subset of a set B if A is "contained" inside B. Notice that A and B may coincide. The relationship of one set being a subset of another is called inclusion or containment.
..... Click the link for more information.INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is detecting some of the most energetic radiation that comes from space. It is the most sensitive gamma ray observatory ever launched.
..... Click the link for more information.Probability theory is the branch of mathematics concerned with analysis of random phenomena.[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities
..... Click the link for more information.Analysis has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function.
..... Click the link for more information.In mathematics, a σ-algebra (or sigma-algebra) over a set X is a nonempty collection Σ of subsets of X that is closed under complementation and countable unions of its members.
..... Click the link for more information.Real analysis is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and
..... Click the link for more information.In mathematics, a σ-algebra (or sigma-algebra) over a set X is a nonempty collection Σ of subsets of X that is closed under complementation and countable unions of its members.
..... Click the link for more information.In mathematics, measurable functions are well-behaved functions between measurable spaces. Functions studied in analysis that are not measurable are generally considered pathological.
..... Click the link for more information.INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is detecting some of the most energetic radiation that comes from space. It is the most sensitive gamma ray observatory ever launched.
..... Click the link for more information.function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
..... Click the link for more information.In mathematics, a σ-algebra (or sigma-algebra) over a set X is a nonempty collection Σ of subsets of X that is closed under complementation and countable unions of its members.
..... Click the link for more information.In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ (pronounced "positive infinity" and "negative infinity"). These new elements are not real numbers.
..... Click the link for more information.empty set is the unique set which contains no elements. In axiomatic set theory it is postulated to exist by the axiom of empty set. The empty set is also sometimes called the null set
..... Click the link for more information.In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In set theory, there is only one null set, and it is the empty set.
..... Click the link for more information.In mathematics, additivity and sigma additivity of a function defined on subsets of a given set are abstractions of the intuitive properties of size (length, area, volume) of a set.
..... Click the link for more information.countable set is a set with the same cardinality (i.e., number of elements) as some subset of the set of natural numbers. The term was originated by Georg Cantor; it stems from the fact that the natural numbers are often called counting numbers.
..... Click the link for more information.In mathematics, two sets are said to be disjoint if they have no element in common. For example, and are disjoint sets.Explanation
Formally, two sets A and B are disjoint if their intersection is the empty set, i.e.
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In mathematics, a tuple is a finite sequence (also known as an "ordered list") of objects, each of a specified type. A tuple containing n objects is known as an "n-tuple".
..... Click the link for more information.In probability theory, the definition of the probability space is the foundation of probability theory. It was introduced by Kolmogorov in the 1930s. For an algebraic alternative to Kolmogorov's approach, see algebra of random variables.
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Topological spaces are mathematical structures that allow the formalization of concepts such as convergence, connectedness and continuity.
..... Click the link for more information.Analysis has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function.
..... Click the link for more information.Probability theory is the branch of mathematics concerned with analysis of random phenomena.[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities
..... Click the link for more information.In mathematics, a Radon measure, named after Johann Radon, on a Hausdorff topological space X is defined in measure theory to be a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular.
..... Click the link for more information.In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) which generalise normed spaces and metric vector spaces.
..... Click the link for more information.In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous.
..... Click the link for more information.In mathematics, a Radon measure, named after Johann Radon, on a Hausdorff topological space X is defined in measure theory to be a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular.
..... Click the link for more information.monotonic function (or monotone function) is a function which preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
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