Baire function

The Baire classes provide a partial classification of real functions; they do is for the first time by René Louis Baire been erected in his dissertation from 1898, and was conceived as a response to questions put to for the first time by Dini (1878 ) question whether every function an analytical pronounced obtained by crossing of elementary functions has representation. Inspiration for such studies has been the Weierstrass approximation theorem formulated in his realization that every continuous function is the limit of Polynomenfolgen. Baire sets this idea continues, in which he defines the class of all functions, the limit of continuous function are consequences, and calls these functions functions of the first class. Limites of functional consequences of the first class are the second Baire class from the second grade - third grade, etc. The study of the Baire classes was later taken up by Lebesgue, Borel, Hausdorff and Young. The hope that it could be shown by classification of all real functions and all sets of real points the continuum hypothesis, has been an important motivating factor in these studies. This hope has been further enhanced by the services of Hausdorff and Alexandroff in 1916 proof of the continuum hypothesis for Borel sets, which are closely related to the Baire classes. Nowadays we know, however, that a complete analytical classification of real functions and sets of points as the proof of the continuum hypothesis are impossible tasks as well.

Definition

Be a.

The zeroth Baire class is the set of all continuous maps

Defined.

For every countable ordinal is the -th Baire class by

Defined.

A function is called bairesch if it is a member of a Baire class. You say bairesch the type of element if it

Is.

Classification of Young

In the classification by Young the set of functions is defined recursively, the limes are falling consequences - called type functions, as well as the set of functions, the limit of growing consequences are - type functions. This is used in the two cases as a basis of the recursion, the amount of continuous functions. A good way to define the young between classes and to illustrate the relationship between the young between classes and Baire classes, provides the notation of cock:

• With the amount of the functions referred to, the limes are a decreasing sequence of functions from a set of functions,

• - Is the set of functions that limes are an increasing sequence of functions,

• - Is the set of functions that limes are any sequence of functions,

If the set of continuous functions referred to, then corresponds to the term already used to denote the set of Baire functions of the type. The amount of young between function type is in this notation and type: . The young between type functions are continuous and the above type - the lower semi continuous functions.

The following rules apply:

• Falls: and.
• If it is isolated (ie, has an immediate predecessor): and.
• For: and.
• For: and.
• If no immediate predecessor (that is a limit ordinal ).
• For: and.
• For:.
• If a limit ordinal is:
• ( Einschiebungssatz )

Relationship between Baire functions and young between functions:

• .
• And

Because the final rule means that the hierarchy of the young between functions can be defined with the help of the hierarchy of Baire functions.

Connection to the Borel sets

The subsets of the set which are Borel sets can be classified as follows:

• Be the set of closed and open subsets of.
• For any quantity was the name of the set of associations and the amount of averages of countably many elements of.

This is called the multiplicative class. is called the additive class. Every Borel set belongs to at least one of these classes ( with ). A function is called B- measurable class if, for each closed set is the prototype member of the multiplicative class. The B- measurable functions can also be characterized by Lebesgue sets. Be for any amount

Said.

It can be shown that the amount of B- measurable functions of the class is the set.

The Baire -type functions are the B- measurable functions of class for every finite ordinal. The Baire -type functions are the B- measurable functions of class ( set of Lebesgue - Hausdorff ) For every transfinite countable ordinal.,

That phrase can write using the notation introduced above in a very compact form:

It is for the young between functions

Properties

The amount of Baire functions of the type is completed for each countable ordinal with respect to the algebraic operations of addition, multiplication, and division:

It also applies:

Each function with at most countably many Unstetigkeitstellen and any characteristic function of a bounded closed set is a function of at most first class. Example of a function of the second class is the Dirichlet function with their analytical representation:

Constructing examples from higher Baire classes is not trivial. The question of whether the Baire classes are empty, has been answered in 1905 by Lebesgue. He manages to show that none of the Baire classes is empty and that the amount of Baire functions and the continuum are equally powerful. The latter means that there are functions that are in any of the Baire classes. One would have to show an explicit example of such a function can construct a non-measurable quantity in the borel 's sense. Not B- measurable sets are the Vitali sets. They are also examples of non- L- measurable quantities. However, in their definition ( the axiom of choice ) were used.

The amount of Unstetigkeitstellen every Baire function of type is lean. This statement is not correct for any Baire function in general. Counter-example is the Dirichlet function. But there is a set whose complement is lean and which is relatively steady at that amount for every Baire function.

Universal function

An important tool to study the Borel sets and Baire functions constitute the so-called universal functions

The function

Called universal function for the set of functions

If

The function

Is universal function relative to the amount, if

Central role in the proof that the Baire classes and the multiplicative classes for each non-empty, plays the Lebesgue's theorem on the universal function: For every positive countable ordinal a universal function exists

For the amount bairesch.

The respective set of Borel sets is as follows: For each countable ordinal universal function exists relative to the multiplicative class so that

The class B

Application in the theory of integration, see the functions of the so-called Baire class. For every sequence of elements of the set

Was

If there is such a number so as to. Otherwise is

The class is defined as follows

Wherein the amount of the continuous functions with compact support respectively. In the Daniell Lebesgue process the integral is first defined for continuous functions with compact support and then through to the functions of Baire class

Extended. With the help of the theorem of Dini can be shown that this definition correctly (ie not dependent on the choice of the monotone increasing sequence of functions ) is.

Generalizations

The term Baire function can be used for pictures

Between arbitrary metric spaces and define. However, not all the properties of real Baire functions readily to the general Baire functions are transferable. All pictures of the metric space of algebraic numbers to itself include, for example, to the zeroth or the first Baire class. If the set of real numbers, then the completeness and the presence of a non-empty insichdichten core is sufficient so that no Baire classes is empty. Each real-valued B- measurable function is a Baire function. If a countable base has, then any B- measurable function of class limit of B- measurable functions lower classes. Each Baire function is B- measurable. If a Baire function of type and a Baire function of type, then their composition is a Baire function of type.

Sources and Notes

• Measure theory
• Topology