Measure (mathematics)

A degree in mathematics is a function that appropriate subsets of a ground set assigns numbers that can be interpreted as a " measure " for the size of these sets. Both the domain of definition of a measure, ie the measurable quantities, as well as the assignment themselves must meet certain requirements, such as those suggested for example by elementary geometric terms of the length of a line, the area of a geometric figure or the volume of a body.

The branch of mathematics that deals with the design and analysis of dimensions, is the measure theory. The general Maßbegriff goes back to work of Émile Borel, Henri Léon Lebesgue, Johann Radon and Maurice René Fréchet. Here dimensions are always in close connection with the integration of functions and form the basis of modern integral terms (see Lebesgue integral ). Since the axiomatization of probability theory by Andrei Kolmogorov stochastics is another major area of application for degree. There, probability measures are used to assign random events that are seen as subsets of a sample space probabilities.

5.1 Maßerweiterungssatz
5.2 Zero quantities, completion of measures
5.3 Measurements on the real numbers
5.4 Limitation of dimensions
5.5 size
5.6 Measurements with densities
5.7 Product dimensions

6.1 Borel measures and regularity
6.2 Hair cal measure
6.3 Convergence of measures

7.1 integration
7.2 spaces of integrable functions
7.3 Probability Theory
7.4 Statistics
7.5 Financial Mathematics

Introduction and History

The elementary geometrical area assigns planar geometric figures such as rectangles, triangles or circles, that certain subsets of the Euclidean plane, numerical values. Surfactant content can be equal to zero, for example, in the empty set, and also in single dots or in lines. Also, the " value" ( infinity) is eg half-planes or the exterior of circles as area before. However, no negative numbers may appear as surface areas.

Furthermore, the surface area of plane geometric figures has a property that additivity is called: If you divide a figure into two or more parts, such as a rectangle by a diagonal into two triangles, then the area of the original figure is the sum of the areas of the parts. "Disassembly " here means that the parts in pairs must be disjoint ( two parts thus have no common points) and that the union of all components results in the output figure. For the measurement of surface areas of more complicated characters, such as circular faces or faces that are enclosed between the function graph ( that is, for the calculation of integrals ), limit values have to be considered by content area. For this it is important that the additivity will still apply if surfaces are divided into a sequence of pairwise disjoint segments. This property is called countable additivity or σ - additivity.

The importance of σ - additivity for the Maßbegriff was first recognized by Émile Borel, who proved in 1894 that the elementary geometric length has this property. The actual measurement problem formulated and examined Henri Lebesgue in 1902 in his PhD thesis: He constructed a σ - additive measure for subsets of the real numbers ( the Lebesgue measure ), which continues the length of intervals, however, a not for all subsets, but for system of subsets, which he called measurable sets. In 1905, Giuseppe Vitali showed that a consistent extension of the length of the term to all subsets of the real numbers is impossible, so that the measurement problem is not solvable.

As important dimensions, such as just the Lebesgue measure, for all subsets ( ie on the power set ) of the base amount can be defined, appropriate domains for measurements must be considered. The σ - additivity suggests that systems of measurable amounts should be closed with respect to countable set operations. This leads to the requirement that the measurable quantities must form a σ - algebra. This means that the universal set is measurable and complements and countable unions of measurable quantities are again measurable.

In the aftermath Jean Thomas Stieltjes and Johann Radon extended the construction of the Lebesgue measure on more general measures in the -dimensional space, the Lebesgue - Stieltjes measure. Maurice René Fréchet studied from 1915 also measures and integrals on any abstract quantities. In 1933 Andrei Kolmogorov published his textbook Basic concepts of probability theory, in which he uses measure theory to give a rigorous axiomatic justification of probability theory (see also history of probability theory ).

Definition

It is a σ - algebra over a non-empty basic amount. A function is called to measure if the following two conditions are met:

σ - additivity: For any sequence of pairwise disjoint sets from.

If the σ - algebra clear from the context, one speaks also of a degree. For ie the measure of the set. The triple is called a measure space. The pair consisting of the basic amount and the σ - algebra defined on it is called the measuring room or even measurable space. A measure is therefore a defined on a measurement space non-negative σ - additive set function.

The measure is called probability measure (or standardized measure ) if, in addition applies. A measure space with a probability measure is a probability space. More generally, it is called a finite measure. If there are countably many sets whose measure is finite and their union resulted wholly, then a σ - finite (or σ - finite ) is called a measure.

Notes and first examples

So a measure assumes non-negative values of the extended real numbers. For the computation with the usual conventions apply, in addition, it is useful to set.
Since all summands of the series are non-negative, this is either convergent or divergent against.
The requirement that the empty set has dimension zero excludes the uninteresting case for all. In fact, the requirement can be equivalently replaced by the condition that exists with. In contrast, the trivial cases for all ( the so-called zero measure ) and for all ( and ) dimensions in the sense of definition.
For an element is

The mapping that assigns every finite set the number of its elements, ie their thickness, as well as the infinite sets in the value, ie counting measure. The counting measure is a finite measure, if is a finite set, and a σ - finite measure, if it is at most countable.
The -dimensional Lebesgue measure is a measure on the σ - algebra of Lebesgue measurable subsets of. It is uniquely determined by the requirement that the - dimensional hyper- rectangles assigns its volume:

The Hausdorff measure is a generalization of the Lebesgue measure on unnecessary integer dimensions. With his help, the Hausdorff dimension can define a notion of dimension, with the example fractal sets can be examined.

Properties

Calculation rules

Directly from the definition, the following basic rules for computing a measure result:

Finite additivity: For pairwise disjoint sets.
Subtraktivität: and shall apply with.
Monotonicity: For with.
For always applies. With the principle of inclusion and exclusion can this formula finally generalize many quantities in the case of finite dimensions on unions and cuts.
σ - subadditivity: For any sequence of sets from.

Continuity properties

The following continuity properties are fundamental for the approximation of measurable quantities. Follow directly from the σ - additivity.

σ -continuity from below: If for an ascending sequence of sets and then apply.
σ -continuity from above: Includes a descending sequence of sets in and then applies.

Uniqueness theorem

For two dimensions on a common measurement space, the following uniqueness theorem holds:

There was an average of stable producer, that is, it applies and for all, with the following properties:

Then we have.

For finite dimensions with the condition 2 is automatically satisfied. In particular, two probability measures are equal if they agree on a stable average producer of the event algebra.

Linear combinations of dimensions

For a family of measures on the same test room and for non-negative real constants a measure is defined by again. In particular, sums and non-negative multiples of dimensions are also dimensions.

For example, if a countable base set and then is a measure on the power set of the Diracmaßen. Conversely, one can show that in countable base set all dimensions to the power set obtained in this way.

Are probability measures, and non-negative real numbers, then the convex combination is a probability measure again. By convex combination of Diracmaßen obtained discrete probability distributions, generally resulting mixture distributions.

Construction of dimensions

Maßerweiterungssatz

Since the elements of σ - algebras, such as in the borel σ - algebra on between, often can not be explicitly specified, dimensions are often constructed by continuation of set functions. The main tool for this is the Maßerweiterungssatz of Carathéodory. He says that on a lot of ring (called a premeasure ) can be any non -negative σ - additive set function to a measure on the σ - algebra generated by continue. The continuation is unique if the premeasure σ - finite.

For example, make all subsets of which can be represented as a finite union of axis-parallel -dimensional intervals, a lot of ring. The elementary volume content of this so-called figures, the Jordan content, is a premeasure on this lot ring. The figures generated by the σ - algebra is the Borel σ - algebra and the continuation of the Jordan content by Carathéodory gives the Lebesgue - Borel measure.

Null sets, completion of measures

Is a measure and a lot with, then that means null set. It is obvious, also assigned to subsets of a set of measure zero dimension zero. However, such amounts are not necessarily measurable, so again lie in. A measure space in which subsets of null sets are always measurable, is called complete.

Is not complete, so can complete the measure space. This purpose, let the set of all subsets of null sets. Substituting and for and, then is a complete measure space and the restriction of to is.

For example, the completion of the Borel - Lebesgue measure is the Lebesgue measure on the Lebesgue measurable subsets of.

Dimensions on the real numbers

The Lebesgue measure on is characterized in that it assigns its length intervals. Its construction can be generalized using a monotonically increasing function of the Lebesgue - Stieltjes measure, assign the intervals, the " weighted length". If the function is also continuous from the right, it constrains a premeasure is defined on the ring of the finite amount associations such intervals. This can be extended to a measure on the Borel sets of or to its completion by Carathéodory. For example, results for the identity map once again the Lebesgue measure; however, is a piecewise constant step function, we obtain linear combinations of Diracmaßen.

If a right-sided continuous and monotonically increasing function also the conditions

Met, the thus constructed Lebesgue - Stieltjes measure is a probability measure. Its distribution function is the same, that means. Conversely, each distribution function of a probability measure on the above properties. With the help of distribution functions can therefore be also those probability measures on simply represent that are neither discrete nor have a Lebesgue density, such as the Cantor distribution.

Restriction of dimensions

Like any function can be a measure of course on a smaller domain, so limit it to a σ - algebra. For example, is obtained by restriction of the Lebesgue measure on the Borel σ - algebra again the Lebesgue - Borel measure back.

More interesting is a restriction to a smaller base amount: Is a measure space and, then, by

A σ - algebra to define the so-called track - σ - algebra. It holds if and only if and. For this is

A measure to define the restriction ( or trace ) is called on. For example, is obtained by restriction of the Lebesgue measure of the interval for a probability measure on the continuous uniform distribution.

Size

Dimensions can be transformed to another chamber with the help of measurable functions from a measure space. Are and measuring rooms, then that means a measurable function if, for all the archetype is. Is now a measure on, then the function is on with a degree. It is size of below and is often referred to with or.

The behavior of integrals in the transformation of dimensions is described by the transform set. Through image dimensions it is possible in analysis to construct measures on manifolds.

Image dimensions of probability measures are probability measures again. This fact plays an important role in the consideration of probability distributions of random variables in the stochastic.

Dimensions with densities

Dimensions are often constructed as " indefinite integrals " of functions with respect to other dimensions. Is a measure space and a non-negative measurable function, is

For another measure to define. The function is called the density function of with respect to ( a short - density). A common notation is.

The set of Radon Nikodým provides information about which dimensions can be represented by means of densities: Is σ - finite, then this is exactly possible if all null sets of even zero amounts of are.

In the stochastics are the distributions of continuous random variables, such as the normal distribution, often indicated by densities with respect to the Lebesgue measure.

Product dimensions

If you can write a basic set as a Cartesian product and are given on the individual factors dimensions, it can be constructed on their so-called product measure. For two measure spaces and denote the product σ - algebra. This is the smallest σ - algebra that contains all the products and quantity. If and σ - finite, then there exists a unique measure on with

Is called the product measure and is denoted by. Entirely analogously, finally forming many dimensions and products. For example, the Lebesgue - Borel measure is thus obtained on the - dimensional Euclidean space as -fold product of the Lebesgue - Borel measure on the real numbers.

Using the set of Fubini can integrals with respect to a product dimension usually calculated by performing step integrations with respect to the individual measurements. In this way, for example, area and volume calculations can be attributed to the determination of one-dimensional integrals.

In contrast to general dimensions products may be formed under certain conditions, in any probability measures ( even uncountable ). Products of probability spaces model, for example, the independent repetition of random experiments.

Measurements on topological spaces

If the base amount is in addition a topological space, we are interested especially for dimensions that have similar characteristics to the Lebesgue measure or the Lebesgue - Stieltjes measure on the topological space with the standard topology. A simple consideration shows that the Borel σ algebra is generated not only on the amount of dimensional intervals, but also of the open portions. Is therefore a Hausdorff space with topology (ie, the amount of open sets ), so we define the Borel σ - algebra as

So as the smallest σ - algebra that contains all open sets. Of course, then, in particular also contains all closed sets and all the quantities which can be written as countable unions or intersections closed or open sets ( cf. Borel hierarchy).

Borel measures and regularity

A measure on a measurement space, Hausdorff space and the Borel σ -algebra is called Borel measure if it is locally finite. That is, each has an open environment whose dimension is finite. If, in addition locally compact, this is equivalent so that all compact sets have finite measure.

A Radon measure is a Borel measure that is on the inside regularly, this means that, for every

Is a Radon measure in addition regularly from the outside, that is, for each valid

So it is called a regular Borel measure.

Many important Borel measures are regular, there are amongst which, the following regularity:

If a locally compact Hausdorff space with a countable base (second countable ), then each Borel measure is regular on.
Every Borel measure on a Polish space is regular.

Probability measures on Polish spaces play an important role in many existential questions of probability theory.

Hair cal measure

The -dimensional Euclidean space is not only a locally compact topological space, but even a topological group with respect to the usual vector addition as a link. The Lebesgue measure also respects this structure in the sense that it is invariant with respect to translations: For all Borel sets and all considered

The concept of Haar measure generalizes this translation invariance on linksinvariante Radon measures on Hausdorff locally compact topological groups. Such a measure always exists and is uniquely determined up to a constant factor. The Haar measure is just finally, if the group is compact; in this case, it may be so normalized to a probability.

Haar measures play a central role in the harmonic analysis are transferred to the methods of Fourier analysis on general groups.

Convergence of measures

The most important notion of convergence for sequences of finite dimensions is weak convergence, which can be defined by means of integrals as follows: It is a metric space. A sequence of finite dimensions on is called weakly convergent to a finite measure, in characters, if for all bounded continuous functions

The Portemanteau theorem gives some other conditions which are the weak convergence of equivalent dimensions. For example, if and only if

Valid for all Borel sets with, with the topological boundary of designated.

The weak convergence of probability measures has an important application in the convergence in distribution of random variables, as occurs in the central limit theorem. Weak convergence of probability measures may be examined with the aid of characteristic features.

Another significant for applications question is when you can choose from consequences of measures weakly convergent sequences, ie, can be characterized as relatively sequentially compact sets of measurements. By the theorem of Prokhorov a class of finite measures on a Polish space is relatively sequentially compact if and only if it is limited and tight. Boundedness means here that is and tautness that, for every compact set with one for all.

Applications

Integration

The term of measurement is closely related to the integration of functions. Modern integral terms as the Lebesgue integral and its generalizations are mostly developed from a measure theoretic basis out. The fundamental relation is the equation

For those in which a measure space is predetermined and the indicator function of the measurable quantity referred, so the function of and usual using the desired linearity and monotonicity properties, the integration can be stepwise initially to simple functions, then a non -negative measurable features and eventually extend to all real - or complex-valued measurable functions. The latter are called integrable and its integral is called ( generalized ) Lebesgue integral with respect to the measure or short integral.

This integral term is a strong generalization of classical integral concepts such as the Riemann integral, because it allows the integration of functions on arbitrary measure spaces. Again, this is in the stochastics of great importance: There corresponds to the integral of a random variable with respect to a given probability measure its expected value.

However, also result for real functions of a real variable advantages over the Riemann integral. Here are primarily the convergence properties at interchange of limit education and integration mentioned, which are described for example by the set of the monotone convergence and the rate of the dominated convergence.

Spaces of integrable functions

Spaces of integrable functions play an important role as the standard spaces of functional analysis. The set of all measurable functions on a measure space, the meet, and are therefore integrable forms a vector space. by

Defines a semi-norm. If we identify functions from this space with each other if they differ only on a null set from each other, one arrives at a normed space. A similar construction can be carried out with general functions for which is a - integrable, and so find the Lp- spaces with the norm

A key result to which the importance of these spaces is due in applications is their completeness. So you are for all Banach spaces. In the important special case arises out even as the norm induced by an inner product; It is therefore in a Hilbert space.

Entirely analogously define complex-valued functions, spaces. Complex spaces are also Hilbert spaces; they play a central role in quantum mechanics, where states of particles are described by elements of a Hilbert space.

Probability Theory

In probability theory, probability measures are used to assign probabilities of random events. Random experiments are described by a probability space, ie, by a measure space whose measure satisfies the additional condition. The basic amount of the sample space that contains the different results that can deliver the experiment. The σ - algebra consists of the events, where the probability measure between numbers and assigns.

Even the simplest case of a finite sample space with the power set as σ - algebra and defined by equal distribution has numerous applications. He plays in elementary probability theory a central role for the description of Laplace experiments, such as the throwing of a cube and drawing from an urn in which all results are as equally likely to be accepted.

Probability measures are often generated as distributions of random variables, ie as image dimensions. Important examples of probability measures on the binomial and Poisson distribution and the geometric and hypergeometric distribution. The probability measures on with Lebesgue density increases - partly because of the central limit theorem - the normal distribution a prominent position. Further examples are the steady uniform distribution or gamma distribution, comprising a number of other distributions, such as the exponential distribution as a special case.

The multidimensional normal distribution is also an important example of probability measures on the -dimensional Euclidean space. Still more general measure spaces play a role in the construction of stochastic processes, such as the Wiener measure on a suitable function space to describe the Wiener process ( Brownian motion ), which occupies a central position in the stochastic analysis in modern probability theory.

Statistics

The basic task of mathematical statistics is to come due to random sampling of observation results to statements about the distribution of characteristics in a population (so-called inferential statistics ). According to a statistical model contains not only a single known as supposed probability measure as a probability space, but a whole family of probability measures on a common measurement space. An important special case of the parametric filters is standard models, which are characterized in that the parameters, vectors, and all have a density with respect to a common measure.

From the observation of will now be closed on the parameters and thus the measure. This is done in classical statistics in the form of point estimators that are constructed using estimators, or with confidence intervals that contain the unknown parameter with a given probability. Using statistical tests hypotheses can also be tested on the unknown probability measure.

In contrast, the distribution parameters are not modeled as an unknown, but even as random in the Bayesian statistics. For this purpose, on the basis of an assumed a priori distribution, determined by means of the additional information obtained by the observation results of an a posteriori distribution of the parameter. These distributions generally are probability measures on the parameter space; However, for a priori distributions under certain circumstances also overall dimensions in question (so-called improper a priori distributions ).

Financial Mathematics

Modern financial mathematics which uses methods of probability theory, stochastic processes, in particular, to model the time evolution of the prices of financial instruments. One central issue is the calculation of fair prices of derivatives.

Typically this is the consideration of various probability measures on the same measurement space: In addition to the real, as determined by the risk appetite of market participants measure of risk-neutral measures are used. Fair prices are then obtained as expectation values of discounted payoffs with respect to a risk-neutral measure. In arbitagefreien and perfect market models this existence and uniqueness risk-neutral measure is ensured.

As can be simple time - and cost- discrete models already analyzed with elementary probability theory, are needed especially in continuous models such as the Black-Scholes model and its generalizations modern methods of martingale theory and stochastic analysis. It can be used as a risk-neutral equivalent martingale measure. These are probability measures having regard to the real -risk measure and a positive density for which the discounted price process is a martingale (or more generally a local martingale ) is. Here the important point is, for example, the set of Girsanow that describes the behavior of Wiener processes with a change of measure.

Generalizations

The concept of the measurement allows many generalizations in different directions. A measure for the purposes of this article is therefore sometimes more specifically called for clarification in the literature positive measure or σ - additive positive measure.

By weakening as required by the definition properties are obtained functions that are considered in the measure theory as precursors of moderation. The general concept is that of a ( non-negative ) set function, ie a function. Quantities of a system of sets over a ground set of values assigns, being usually still required that the empty set gets the value zero A table of contents is a finitely additive set function; a σ - additive content is called premeasure. The Jordan - content on the Jordan - measurable subsets of is an example of using an additive set function, with which is not σ - additive. A measure is therefore a premeasure whose domain is a σ - algebra. Outer dimensions, so lot of features that are monotone and σ - subadditive, represent an important intermediate in the construction of dimensions from Prämaßen by Carathéodory is: A premeasure on a lot of ring is initially continued to an outer measure on the whole power set whose restriction to measurable quantities gives a measure.

Other types of generalizations of Maßbegriffs obtained if one abandons the requirement that the values must be in, but retains the remaining properties. A signed measure negative values are allowed, so it can take values in the interval (alternatively ). For a complex number as a range of values is called a complex measure. The value here is, however, not allowed, ie, a positive measure is indeed always a signed measure, but only finite dimensions can also be interpreted as a complex dimensions. In contrast to the positive dimensions form the signed and the complex dimensions on a measurement space to a vector space. Such spaces play an important role as dual spaces of spaces of continuous functions by the Riesz representation theorem. Signed and complex dimensions can be written as linear combinations of positive measures by the decomposition theorem of Hahn and Jordan. Also the set of Radon Nikodým remains valid for them.

An even further generalization provide measurements represent values in any Banach spaces, the so-called vectorial dimensions. Dimensions on the real numbers whose values are orthogonal projections of a Hilbert space, so-called spectral measures are used in the spectral theorem for the representation of self-adjoint operators, which among other things in the mathematical description of quantum mechanics plays an important role (see also Positive Operator Valued Probability Measure). Mass orthogonal values are Hilbert space -quality measurements, where the measurements of disjoint sets are orthogonal to one another. With the help of spectral representations can be given of stationary time series and stationary stochastic processes.

Random measurements are random variables whose values are measures. They are used for example in stochastic geometry for the description of random geometric structures. For stochastic processes whose paths have discontinuities, such as the Lévy processes, the distributions of these jumps can be represented by random Zählmaße.

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