Product dimension

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In mathematics, a product measure is a special measure on the product of measure spaces. It is characterized by the fact that it assigns the product of the measures of the individual sets to a Cartesian product of sets. So the dimensional Lebesgue Borel measure on the straight the fold product measure of the one-dimensional Lebesgue Borel measure. In probability theory , products of probability measures are used to model stochastic independence .

Construction of the product dimension

introduction

If you (the to the usual real line - and axis) with the one-dimensional Lebesgue measure thinks it is obvious a measure at the level of defining such that for measurable amounts applies

This then results in particular for the two-dimensional dimension of a rectangle

the formula , the well-known formula according to which the area of ​​a rectangle is equal to the product of its side lengths.

Since even the simplest geometric figures , such as triangles or circles, cannot be represented as Cartesian products, the set function still has to be continued to a measure on a σ-algebra .

Products of two dimensions

For any two measurement spaces and , the product σ-algebra must first be defined. This is that of the product of and

generated algebra, i.e. the smallest algebra that contains. (This step is necessary because the product itself is generally not an -algebra, but just a half-ring .)

Now and are two dimensional spaces. Analogously to the example above, one would like to define a measure on the product σ-algebra that satisfies for all . A measure that fulfills this condition is then called a product measure . Such a measure always exists, as can be shown, for example, with Carathéodory's measure extension theorem . However, such a measure is not necessarily clearly determined. If, however, two σ-finite measure spaces are involved, then σ-finite is also and there is exactly one product measure . It is denoted by. In this case, the product dimension can be represented as an integral according to the Cavalieri principle : For applies

Products of finite dimensions

Be with and a family of bespoke spaces. A measure defined on the corresponding product σ-algebra is then called the product measure of if for all

applies. The existence of is shown by complete induction over by means of the product of two measures. Analogously to this, one obtains the uniqueness of after the continuation clause if for all is -finite.

According to define with the Produktmaßraum of .

Remarks

  • With the help of this definition the Cavalieri principle can be formulated in its most general form on the Lebesgue-measurable subset for every (almost everywhere) .
  • The theorems of Fubini and Tonelli also apply very generally (i.e. not necessarily only for Euclidean space) to measurable functions under the assumption of σ-finite measure spaces.
  • For the uniqueness statement of it is really necessary that both measure spaces are -finite. If one sets (the Borel σ-algebra restricted to [0,1]) and chooses for the Lebesgue measure , for the non-finite counting measure , there are at least three different product measures, although one of the measure spaces is still -finite.
  • The product measure of two complete measures is generally not complete again, e.g. for every subset there is a - zero set , but this set is only in for , i.e. H. it applies
  • In contrast, in Borel's σ-algebra applies to all .
  • If and are two probability spaces that each describe a random experiment , then the product models the joint experiment, which consists of performing the two individual experiments independently of one another.

Infinite product dimensions

In probability theory , one is particularly interested in the existence of infinite product measures , i.e. in products of countable or uncountable many probability measures . These enable the examination of limit values ​​or important constructions such as those of independently identically distributed random variables or product models in stochastics and statistics .

definition

Both definitions fall back on the constructions of the finite product dimension.

Countable index set

For a countably infinite index set , here as an example , the above product formula can no longer be formulated explicitly. Instead, one demands that it hold for the first probability measures, and this for any . So are probability spaces for if so, the probability measure on

the product measure of , if for all and all that

is.

Uncountable index set

For an uncountably infinite index set , the above procedure reaches its limits, since a definition using the first measures is no longer useful. Instead, one considers projections of a probability measure from the uncountable product space onto the finite product spaces. The image measure under such a projection should then agree with the finite product of the probability measures.

So are now probability spaces for given and is

the uncountable product space and

the projection onto the components . Then a probability measure is called the product measure if for every finite subset the image measure coincides with the finite product measure of . So it should

be valid. In particular, the definition for countable products is a special case of this definition with .

Existence and uniqueness

The Andersen-Jessen theorem provides both the existence of a product measure and its uniqueness . There is a wide variety of evidence for the existence of product dimensions, which differ according to the degree of their generality and their requirements. For example, there are separate sentences about the existence of a product measure in the case of an infinitely repeated coin toss. Andersen-Jessen's theorem provides the existence and uniqueness for any index sets and without making any special requirements, and thus answers the question in its entirety.

Demarcation

Product dimensions should not be confused with dimensions on a product space . These are used in the theory of stochastic processes and differ from the product dimensions in particular in that the above product formulas, which correspond to stochastic independence , no longer have to apply. A typical example for this would be a Markov process : The question arises whether a probability measure exists on the product of the state space that describes the process as a whole. This probability measure is then certainly not a product measure in the above sense, since Markov processes are characterized by their dependence and accordingly the above product formulas will not apply.

Important of these existence theorems for dimensions on product spaces are the Ionescu-Tulcea theorem and Kolmogorov's extension theorem . The first provides the existence of a probability measure which is defined by means of Markov kernels , the second the existence of a measure with predetermined marginal distributions which are determined by means of projective families of probability measures . Both sets can also be used as special cases for the construction of product dimensions. However, they do not give results as general as the Andersen-Jessen Theorem. For example, Kolmogorov's extension kit only applies to Borel measuring rooms .

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