Baire classes

Families of real functions which are defined inductively using the ordinal number of limit transitions involved in the definition of the function, and which constitute the classification of functions proposed in 1899 by R. Baire [1], known as the Baire classification. The set of all continuous functions , where is a metric space, is called the zero-th Baire class . The first Baire class is the set of discontinuous functions which are the limit of a sequence of continuous functions converging at each point. The Baire class , where is an ordinal number of the first or second class, is defined as the set of functions which do not form part of any of the preceding classes, but which may be represented as , where , . The union of the Baire classes over all ordinal numbers of the first and second class constitutes the set of Baire functions. This is the minimal class of functions containing all the continuous functions and closed with respect to pointwise convergence. A linear combination, product and quotient (for non-zero denominators) of functions of a Baire class not higher than is a function in a Baire class not higher than . A uniformly convergent sequence of functions of a Baire class not higher than has as limit a function in a Baire class not higher than . Necessary and sufficient conditions have been established for a sequence of functions in a Baire class not higher than to converge to a function in a Baire class not higher than [4]. The union of all dense-in-itself sets is called the kernel of the topological space . If is a complete space with a non-empty dense-in-itself kernel, none of the Baire classes is empty [2]. The set of Baire functions coincides with the set of Borel-measurable functions (cf. Borel measure), and for this reason they are all Lebesgue-measurable (cf. Lebesgue measure). A Lebesgue-measurable function is equivalent to a Baire function of a Baire class not higher than two [3]. Baire, who considered functions defined on (mainly on ), made the most detailed study of functions of the first class. He showed that a necessary and sufficient condition for a discontinuous function to belong to the first class is the existence of a point of continuity of the induced function on each perfect set (Baire's theorem). This theorem is applicable to a function if has the Baire property [2]. The concept of a Baire function can be naturally generalized to functions , where is an arbitrary metric space.

References

[1]	R. Baire, "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars (1905)
[2]	F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978)) MR1034865 MR0979016 MR0031025 Zbl 1175.01034 Zbl 45.0123.01
[3]	I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian) MR0640867 MR0409747 MR0259033 MR0063424 Zbl 0097.26601
[4]	B.M. Gagaeff, "Sur les suites convergentes de fonctions mesurables " Fund. Math. , 18 (1932) pp. 182–188 Zbl 0004.20501

Comments

A subset of a topological space is called dense-in-itself if it has no relatively-isolated points. A modern English reference for the notion of Baire classes is [a1]. For an introduction to the topic and some applications, cf. [a2].

References

[a1]	A.C.M. van Rooy, W.H. Schikhof, "A second course on real functions" , Cambridge Univ. Press (1982) MR655599
[a2]	R.P Boas jr., "A primer of real functions" , Math. Assoc. Amer. (1981) MR0652482 Zbl 0473.26002