Bairean class

The Baier classes represent a partial classification of the real functions .

It was set up for the first time by René Louis Baire in his dissertation in 1898 and was conceived as an answer to the question posed for the first time by Dini (1878) whether every function has an analytical representation, i.e. a representation obtained from elementary functions by crossing borders . The inspiration for such investigations was the knowledge formulated by Karl Weierstrass in his approximation theorem that every continuous function is a limit of polynomial sequences. Baire continues this idea by defining the class of all functions that are limits of continuous function sequences and calling these functions functions of the first class . Limites of sequences of functions from the first class form the second Baier class, from the second class - the third class etc. The study of the Baier classes was later taken up by Henri Léon Lebesgue , Émile Borel , Felix Hausdorff and William Henry Young . The hope that one could prove the continuum hypothesis by classifying all real functions and all sets of real points has been an important motivating factor in these investigations. This hope has been reinforced by the proof of the continuum hypothesis for Borel sets , which are closely related to the Baier classes, provided by Hausdorff and Pawel Sergejewitsch Alexandrow in 1916 . Today, however, we know that a complete analytical classification of the real functions and point sets, like the proof of the continuum hypothesis, are unsolvable problems.

definition

Be for one . ${\displaystyle A={\underset {i=1}{\overset {n}{\mathbf {\times } }}}[a_{i},b_{i}]\subset \mathbb {R} ^{n}}$ ${\displaystyle n\leq \aleph _{0}}$

The zeroth Baier class on is called the set of all continuous mappings ${\displaystyle H_{0}}$ ${\displaystyle A}$

${\displaystyle f\colon A\to \mathbb {R} }$

Are defined.

For each maximum countable ordinal is the th Baire function on by ${\displaystyle \kappa >0}$ ${\displaystyle \kappa }$ ${\displaystyle A}$

${\displaystyle H_{\kappa }=\left\{f:\ f\notin C^{(<\kappa )}=\bigcup _{\alpha <\kappa }H_{\alpha };\ \exists (f_{n})_{n=1,2,...}\left(\forall n\left(f_{n}\in C^{(<\kappa )}\right)\land f=\lim _{n\to \infty }f_{n}\right)\right\}}$

Are defined.

A function is called Bairean if it is a member of a Bairean class. It is called bairesch of the type when it is element of ${\displaystyle \kappa }$

${\displaystyle C^{(\kappa )}=C^{(<\kappa )}\cup H_{\kappa }}$

is.

Classification by Young

In Young's classification, the set of functions is recursively defined that are limits of decreasing sequences - called functions of type , and the set of functions that are limits of increasing sequences - functions of type . In both cases, the recursion is based on the set of continuous functions. Hahn's notation offers a good way of defining the Young classes and illustrating the connection between the Young classes and the Baie classes : ${\displaystyle g_{\xi }}$ ${\displaystyle G_{\xi }}$

• with denotes the set of functions that are limits of a decreasing sequence of functions from a function set , ${\displaystyle S_{1}}$ ${\displaystyle S}$

• ${\displaystyle S^{1}}$- is the set of functions which are limes out of a growing sequence of functions , ${\displaystyle S}$

• ${\displaystyle S^{*}}$- is the set of functions that are limits of any sequence of functions , ${\displaystyle S}$

• ${\displaystyle S^{(0)}=S_{0}=S^{0}=S}$

• ${\displaystyle S^{(<\xi )}=\cup _{\alpha <\xi }S^{(\alpha )}}$

• ${\displaystyle S^{(\xi )}={\left(S^{(<\xi )}\right)}^{*}}$

• ${\displaystyle S^{<\xi }=\cup _{\alpha <\xi }S^{\alpha }}$

• ${\displaystyle S^{\xi }={\left(S^{<\xi }\cup S_{<\xi }\right)}^{1}}$

• ${\displaystyle S_{\xi }={\left(S^{<\xi }\cup S_{<\xi }\right)}_{1}}$

• ${\displaystyle S_{\xi }^{\xi }=S^{\xi }\cap S_{\xi }}$

If the set of continuous functions designates, then the designation corresponds to the designation already used for the set of Bairean functions of the type . The amount of Young's function of type is in this notation and type : . The Young's type functions are the above steady and type - the under semi-continuous functions. ${\displaystyle C}$ ${\displaystyle C^{(\xi )}}$ ${\displaystyle \xi }$ ${\displaystyle g_{\xi }}$ ${\displaystyle C_{\xi }}$ ${\displaystyle G_{\xi }}$ ${\displaystyle C^{\xi }}$ ${\displaystyle g_{1}}$ ${\displaystyle G_{1}}$

The following rules apply:

• If : and . ${\displaystyle \eta <\xi }$ ${\displaystyle C^{\eta }\subseteq C^{\xi },\ C^{\eta }\subseteq C_{\xi },\ C_{\eta }\subseteq C^{\xi },\ C_{\eta }\subseteq C_{\xi },\ C^{\eta }\cup C_{\eta }\subseteq C_{\xi }^{\xi },\ C_{\eta }^{\eta }\subseteq C_{\xi }^{\xi }}$ ${\displaystyle C^{(\eta )}\subseteq C^{(\xi )}}$
• If is isolated (has an immediate predecessor ): and . ${\displaystyle \xi }$ ${\displaystyle \xi -1}$ ${\displaystyle C^{<\xi }=C^{\xi -1}\,}$ ${\displaystyle C_{<\xi }=C_{\xi -1}\,}$
• For : and . ${\displaystyle \xi >0}$ ${\displaystyle (C^{\xi })_{1}=C_{\xi +1}\,}$ ${\displaystyle (C_{\xi })^{1}=C^{\xi +1}\,}$
• For : and . ${\displaystyle \xi >1}$ ${\displaystyle (C_{<\xi })^{1}=C^{\xi }\,}$ ${\displaystyle (C^{<\xi })_{1}=C_{\xi }\,}$
• If no immediate predecessor (that is a limit ordinal) . ${\displaystyle \xi }$ ${\displaystyle C_{<\xi }\cup C^{<\xi }=C^{<\xi }=C_{<\xi }\,}$
• For : and . ${\displaystyle \xi >0}$ ${\displaystyle (C_{\xi }^{\xi })_{1}=C_{\xi }\,}$ ${\displaystyle (C_{\xi }^{\xi })^{1}=C^{\xi }\,}$
• For : . ${\displaystyle \xi >0}$ ${\displaystyle (C_{\xi }^{\xi })^{*}=C_{\xi +1}^{\xi +1}\,}$
• If a limit number is: ${\displaystyle \xi }$ ${\displaystyle \cup _{\alpha <\xi }C_{\alpha }^{\alpha }=C_{<\xi }\cup C^{<\xi }\,}$
• ${\displaystyle \forall (f_{1}\in C_{\xi })\ \forall (f_{2}\in C^{\xi })\ \exists (f\in C_{\xi }^{\xi })\ ((f_{1}\leq f_{2})\Rightarrow (f_{1}\leq f\leq f_{2}))}$ ( Insert sentence ),

Relationship between Baier functions and Young functions:

• ${\displaystyle C^{\xi }\cup C_{\xi }\subseteq C^{(\xi )}=C_{\xi +1}^{\xi +1}=C^{\xi +1}\cap C_{\xi +1}.\,}$.
• ${\displaystyle C^{\xi }=\left(C^{(<\xi )}\right)^{1}\,}$ and ${\displaystyle C_{\xi }=\left(C^{(<\xi )}\right)_{1}\,}$

Because of the last rule means that the hierarchy of the Young's functions can also be defined with the help of the hierarchy of the Bairean functions. ${\displaystyle C^{(0)}=C^{0}=C_{0}=C}$

Connection to the Borel sets

The subsets of the set , which are Borel sets, can be classified as follows: ${\displaystyle A}$

• ${\displaystyle F}$be the set of closed and open subsets of . ${\displaystyle G}$ ${\displaystyle A}$
• ${\displaystyle M^{\{1\}}}$for an arbitrary set let the notation of the set of unions and the set of averages of countably many elements of . ${\displaystyle M}$ ${\displaystyle M_{\{1\}}}$ ${\displaystyle M}$
• ${\displaystyle (F\cup G)^{\{<\xi \}}=\bigcup \nolimits _{\eta <\xi }(F\cup G)^{\{\eta \}}}$
• ${\displaystyle (F\cup G)_{\{<\xi \}}=\bigcup \nolimits _{\eta <\xi }(F\cup G)_{\{\eta \}}}$
• ${\displaystyle (F\cup G)^{\{\xi \}}=\left((F\cup G)^{\{<\xi \}}\cup (F\cup G)_{\{<\xi \}}\right)^{\{1\}}}$
• ${\displaystyle (F\cup G)_{\{\xi \}}=\left((F\cup G)^{\{<\xi \}}\cup (F\cup G)_{\{<\xi \}}\right)_{\{1\}}}$
• ${\displaystyle B^{\{1\}}=G,\ B_{\{1\}}=F}$
• ${\displaystyle B^{\{1+\xi \}}=(F\cup G)^{\{\xi \}},\ B_{\{1+\xi \}}=(F\cup G)_{\{\xi \}}\,}$

One calls the multiplicative class . is called the additive class . Every Borel set belongs to at least one of these classes . A function is called B-measurable of the class if the archetype is an element of the multiplicative class for every closed set . The B-measurable functions can also be characterized by Lebesgue sets. Be for any amount ${\displaystyle (F\cup G)_{\{\xi \}}}$ ${\displaystyle \xi }$ ${\displaystyle (F\cup G)^{\{\xi \}}}$ ${\displaystyle \xi }$ ${\displaystyle \xi <\Omega }$ ${\displaystyle \xi }$ ${\displaystyle U}$ ${\displaystyle f^{-1}(U)}$ ${\displaystyle \xi }$ ${\displaystyle M}$

• ${\displaystyle (*,M)=\{f:\ \forall y[f\geq y]\in M\}}$
• ${\displaystyle (M,*)=\{f:\ \forall y[f\leq y]\in M\}}$
• ${\displaystyle (M,M)=(M,*)\cap (*,M)}$

whereby . ${\displaystyle [f\;R\;y]=\{x:\ f(x)\;R\;y\}}$

It can be shown that the set of B-measurable functions of the class is the set . ${\displaystyle \xi }$ ${\displaystyle ((F\cup G)_{\{\xi \}},(F\cup G)_{\{\xi \}})}$

For any finite ordinal number , the Bairean functions of the type are the B-measurable functions of the class . For every transfinite ordinal number that can at most be counted, the Baier functions of the type are the B-measurable functions of the class ( Lebesgue-Hausdorff theorem ). ${\displaystyle \xi }$ ${\displaystyle \xi }$ ${\displaystyle \xi }$ ${\displaystyle \xi }$ ${\displaystyle \xi +1}$

This sentence can be written down in a very compact form using the notation introduced above:

${\displaystyle C^{(\xi )}=(B_{\{\xi +1\}},B_{\{\xi +1\}})\,}$.

It reads for the Young functions

${\displaystyle C^{\xi }=(B_{\{\xi \}},*),\ C_{\xi }=(*,B_{\{\xi \}})\,}$.

properties

The set of Bairean functions of the type is closed for every ordinal number that can at most be counted with respect to the algebraic operations of addition, multiplication and division: ${\displaystyle \xi }$ ${\displaystyle \xi }$

• ${\displaystyle \{f,g\}\subset C^{(\xi )}\Rightarrow \{f+g,fg\}\subset C^{(\xi )}}$
• ${\displaystyle \{f,g\}\subset C^{(\xi )}\land \forall x(|g(x)|>0)\Rightarrow {\frac {f}{g}}\in C^{(\xi )}}$

It also applies:

• ${\displaystyle \{f,g\}\subset C^{(\xi )}\Rightarrow \{|f|,\ \max(f,g),\ \min(f,g)\}\subset C^{(\xi )}.}$
• ${\displaystyle \{f_{n}\}_{n\in \mathbb {N} }\subset C^{(\xi )}\Rightarrow {\underset {n\to \infty }{\mathrm {Lim} }}f_{n}\in C^{(\xi )}}$

Every function with at most a countable number of discontinuities as well as every characteristic function of a bounded closed set is a function of at most the first class. An example of a function of the second class is the Dirichlet function with its analytical representation:

${\displaystyle D(x)=\lim _{m\to \infty }\lim _{n\to \infty }\cos ^{2n}m!\pi x.}$

Constructing examples from higher Baier classes is not trivial. The question of whether the Baierian classes are empty was answered by Lebesgue in 1905. He succeeds in showing that none of the Bairean classes is empty and that the set of Bairean functions and the continuum are equally powerful. The latter means that there are functions that are not in any of the Baier classes. In order to be able to show an explicit example for such a function, one would have to construct a set that cannot be measured in the Borelian sense. Quantities that cannot be B-measured are the Vitali quantities . They are also an example of quantities that cannot be measured by L. However, ( the axiom of choice ) is used in their definition . ${\displaystyle \mathrm {AC} }$

The set of discontinuities of any Baier function of type is lean . This statement is generally incorrect for any Baire function. The opposite example is the Dirichlet function . For every Baier function, however, there is a set whose complement is lean and for which is continuous relative to this set. ${\displaystyle 1}$ ${\displaystyle f}$

Universal function

The so-called universal functions are an important instrument for investigating the Borel sets and the Baier functions .

The function

${\displaystyle F:A\times [0,1]\to \mathbb {R} }$

is called universal function for the set of functions

${\displaystyle C\subset \{f|f\colon A\to \mathbb {R} \}}$,

if

${\displaystyle \forall f\in C\ \exists t\in [0,1]\ \forall x(F(x,t)=f(x)).}$

The function

${\displaystyle F\colon [0,1]\to M}$

is called the universal function relative to the set , if ${\displaystyle M}$

${\displaystyle \forall X\in M\ \exists t\in [0,1](F(t)=X).}$

Lebesgue's theorem on the universal function plays a central role in the proof that the Baieean classes and the multiplicative classes are not empty for each : for every positive ordinal number that can be counted at most there is a universal function ${\displaystyle H_{\xi }}$ ${\displaystyle \xi \,}$ ${\displaystyle \xi <\Omega }$ ${\displaystyle \xi }$

${\displaystyle F_{\xi }\colon A\times [0,1]\to \mathbb {R} }$

for the crowd that is bairesch. ${\displaystyle C^{(<\xi )}}$

The corresponding theorem for Borel sets reads: For every ordinal number that can be counted at most, there exists a universal function relative to the multiplicative class such that ${\displaystyle \xi }$ ${\displaystyle F_{\xi }}$ ${\displaystyle \xi }$

${\displaystyle \{(x,z):\ x\in F_{\xi }(z)\}\in (F\cup G)_{\{\xi \}}\subset {\mathcal {P}}(A\times [0,1]).}$

The class B ⁺

The functions of the so-called Baie class are used in integration theory . For each sequence of elements of the set ${\displaystyle B^{+}}$ ${\displaystyle (a_{n})_{n=1,2,...}}$

${\displaystyle {\overline {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,+\infty \}}$

be

${\displaystyle {\underset {n}{\textrm {Sup}}}\;a_{n}={\underset {n}{\textrm {sup}}}\;a_{n},\,}$

if there is such a number such that . Otherwise be ${\displaystyle a\in \mathbb {R} \cup \{-\infty \}}$ ${\displaystyle \forall n(a_{n}\leq a)}$

${\displaystyle {\underset {n}{\textrm {Sup}}}\;a_{n}=+\infty .}$

The class is defined as follows ${\displaystyle B^{+}}$

${\displaystyle B^{+}=\left\{f={\underset {n}{\textrm {Sup}}}\;f_{n}:\ \ \forall m(f_{m}\leq f_{m+1}\land f_{m}\in C_{c}(\mathbb {R} ^{n}))\right\},}$

where denotes the set of continuous functions with compact support . In the Daniell Lebesgue process, the integral first for continuous functions with compact support defined and then the functions of the Baire class by ${\displaystyle C_{c}(\mathbb {R} ^{n})}$ ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ ${\displaystyle B^{+}}$

${\displaystyle \int _{\mathbb {R} ^{n}}f(x)\operatorname {d} x={\underset {n}{\textrm {Sup}}}\left(\int _{\mathbb {R} ^{n}}f_{n}(x)\operatorname {d} x\right)}$

extended. With the help of Dini's theorem, it can be shown that this definition is correct (i.e. not dependent on the choice of the monotonically growing sequence of functions ). ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$

Generalizations

The term bairean function can be used for illustrations

${\displaystyle f\colon X\to Y}$

between any metric spaces and define. However, not all properties of the real Bairean functions can be easily transferred to the general Bairean functions. All mappings of the metric space of the algebraic numbers to themselves belong, for example, to the zeroth or the first Baier class. If the set of real numbers is, then the completeness and the presence of a non-empty in- self-dense kernel of is sufficient for none of the Baier classes to be empty. Every real-valued B-measurable function is a Baier function. If has a countable basis, then every B-measurable function of the class Limes is of B-measurable functions of lower classes. Every Baier function is B-measurable. If a Baier function is of type and a Baier function is of type , then its composition is a Baier function of type . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle \xi <\Omega }$ ${\displaystyle Y}$ ${\displaystyle 2<\xi <\Omega }$ ${\displaystyle f}$ ${\displaystyle \alpha }$ ${\displaystyle g}$ ${\displaystyle \beta }$ ${\displaystyle \alpha +\beta }$

Sources and Notes

1. ^ Israel Kleiner : Evolution of the Function Concept: A Brief Survey. In: The College Mathematics Journal. Vol. 20, No. 4, September 1989, ISSN  0746-8342 , pp. 282-300.
2. ↑ This idea can be traced back to Cantor. In his work he shows about infinite linear point manifolds. ( Mathematische Annalen . Vol. 23, 1884, pp. 453-488, digitized ) that every closed set of real points is a union of a perfect and a countable set, and suggests that this scheme can be expanded in such a way so that all sets of real points can be described by simple sets. The works of Cantor and Baire are considered to be the first in the field of so-called descriptive set theory (from the Latin describere "to describe").
3. ↑ Felix Hausdorff : The mightiness of Borel sets. In: Mathematical Annals. Vol. 77, No. 3, 1916, pp. 430-437, digitized .
4. ↑ The restriction to the maximum number of ordinal numbers that can be counted is not absolutely necessary. It is based on the fact that all -th Bairean classes are empty for uncountable -s, which can easily be shown by transfinite induction (see: Péter Komjáth, Vilmos Totik: Problems and Theorems in Classical Set Theory. Springer, New York NY 2006, ISBN 0-387-30293-X .) ${\displaystyle \kappa }$ ${\displaystyle \kappa }$
5. ↑ Л. В. Канторович : Об обобщенных производных непрерывных функций. In: Математический сборник. Vol. 39, No. 4, 1932, ISSN  0321-4540 , pp. 153-170, PDF ( Memento of September 28, 2007 in the Internet Archive ).
6. ↑ Friedrich Hartogs : To the representation and expansion of the Bairean functions. In: Mathematical Journal . Vol. 42, No. 1, December 1937, digitized from digizeitschriften.de.
7. ^ ^{A b} Hans Hahn : Real functions (= mathematics and its applications in monographs and textbooks. Vol. 13, ZDB -ID 503786-4 ). Akademisches Verlagsgesellschaft mbH, Leipzig 1932.
8. ↑ At this point it should be underlined once again that this is a real-valued function in the narrower sense. If one allows that the Limes functions also take on the values ​​± ∞, then every Young function is semi-continuous, but not every semi-continuous function is Young.
9. ↑ All these rules can be found in: Hans Hahn: Real functions (= mathematics and its applications in monographs and textbooks. Vol. 13, ZDB -ID 503786-4 ). Akademisches Verlagsgesellschaft mbH, Leipzig 1932, under 35.1.1, 35.1.11, 35.1.21, 35.1.5, 34.2.1, 34.2.11 and 34.1.1.
10. ↑ ^{a b} Kazimierz Kuratowski : Topology. Volume 1. New edition, revised and augmented. Academic Press et al., New York et al. 1966, § 31.
11. ^ ^{A b} Henri Lebesgue : Sur les fonctions representables analytiquement. In: Journal de Mathématiques Pures et Appliquées. Série 6, Vol. 1, 1905, ISSN  0021-7824 , pp. 139-216, digitized from Gallica .
12. ^ ^{A b} Felix Hausdorff : Set theory (= Göschen's teaching library . Vol. 7, ZDB -ID 503797-9 ). 2nd, revised edition. de Gruyter, Berlin et al. 1927, § 39.
13. ↑ ^{a b} Casper Goffman: Real functions. Bibliographisches Institut-Wissenschafts-Verlag, Mannheim et al. 1976, ISBN 3-411-01510-1 .
14. ↑ Isidor P. Natanson: Theory of the functions of a real variable. Unchanged reprint of the 4th edition. Harri Deutsch, Zurich et al. 1977, ISBN 3-87144-217-8 (also in digital form in Russian at INSTITUTE OF COMPUTATIONAL MODELING SB RAS, Krasnoyarsk ).
15. ↑ Example of a function from can be found in: Hans Hahn: Real functions (= mathematics and its applications in monographs and textbooks. Vol. 13, ZDB -ID 503786-4 ). Akademisches Verlagsgesellschaft mbH, Leipzig 1932, under § 37.4. ${\displaystyle H_{3}}$
16. ↑ One can even show that there is a -th Bearean class for a Baier universal function . For the set of Baier functions of the type, there is no Baier universal function from the -th Baier class. For the set of Young's functions of the type, there is a Young universal function that is also of the type . (see: Л. В. Канторович: Об универсальных функциях. In: урнал Ленинградского физико-математического общества. Vol 2, H. 2, 1929. ZDB -ID 803408-4 , S. 13-21, PDF ( Memento of 28 September 2007 in the Internet Archive )). ${\displaystyle C^{(<\xi )}}$ ${\displaystyle \xi }$ ${\displaystyle \xi }$ ${\displaystyle \xi }$ ${\displaystyle \xi }$ ${\displaystyle \xi }$
17. ^ Eberhard Freitag : Lectures on Analysis. Script, part II, pdf .
18. ↑ Sashi M. Srivastava: A course on Borels sets (= Graduate Texts in Mathematics. Vol. 180). Springer, New York NY et al. 1998, ISBN 0-387-98412-7 .