NDLI: A fine correlation between Baire and Borel functional hierarchies

Content Provider	Springer Nature Link
Author	Zakharov, Valeriy K. Rodiov, Timofey V.
Copyright Year	2013
Abstract	There are two widely known functional hierarchies on a topological space $(T,\mathcal {G})$ . The transfinite chain $A(T)\subset\dots\subset \operatorname{Lim}_{\alpha}A(T) \subset\dots\subset \operatorname{Lim}_{\omega_{1}}A(T)$ , where A(T) is an initial family of functions on T and $\operatorname{Lim}_{\alpha}A(T)$ consists of all pointwise limits of sequences of functions from preceding classes, is called the Baire convergence hierarchy. The transfinite chain $M(T,\mathcal {B}_{0})\subset\dots\subset M(T,\mathcal {B}_{\alpha})\subset\dots\subset M(T,\mathcal {B}(T,\mathcal {G}))$ , where $\mathcal {B}_{0}\subset \dots \mathcal {B}_{\alpha}\subset \dots \subset \mathcal {B}(T,\mathcal {G})$ is some hierarchy in the σ-algebra $\mathcal {B}(T,\mathcal {G})$ of Borel sets and $M(T,\mathcal {B}_{\alpha})$ is the family of all $\mathcal {B}_{\alpha}$ -measurable functions, is called the Borel descriptive hierarchy. There are two famous correlations between these hierarchies. The first one is the Lebesgue–Hausdorff correlation with the initial family $C(T,\mathcal {G})$ and Young–Hausdorff ensembles $\mathcal {B}_{0}\equiv \mathcal {G}$ , $\mathcal {B}_{1}\equiv \mathcal {F}_{\sigma}$ , $\mathcal {B}_{2}\equiv \mathcal {G}_{\delta\sigma}$ , $\mathcal {B}_{3}\equiv \mathcal {F}_{\sigma\delta\sigma}$ , …, which is valid only for perfectly normal spaces. The second one is the Banach correlation with the initial family $M(T,\mathcal {F}_{\sigma})$ , which is valid only for perfect spaces.For an arbitrary topological space $(T,\mathcal {G})$ there is the general correlation with the initial family $M(T,\mathcal {G}_{\lambda})$ , $\mathcal {B}_{0}\equiv \mathcal {G}$ , $\mathcal {B}_{1}\equiv$ $\mathcal {G}_{\lambda}$ , $\mathcal {B}_{2}\equiv \mathcal {G}_{\lambda\lambda}$ , where $\mathcal{E}_{\lambda}\equiv \left\{\bigcup (E_{n}\cap(T\setminus H_{n})\mid n\in\omega)\mid E_{n},H_{n}\in\mathcal{E}\right\}$ .In this paper we establish the fine Baire–Borel correlation, i. e., we find the initial family of uniform functions strictly intermediate between $C_{b}(T,\mathcal {G})$ and $M_{b}(T,\mathcal {G}_{\lambda})$ .
Ending Page	402
Page Count	19
Starting Page	384
File Format	PDF
ISSN	02365294
e-ISSN	15882632
Journal	Acta Mathematica Hungarica
Issue Number	2
Volume Number	142
Language	English
Publisher	Springer Netherlands
Publisher Date	2013-09-18
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Baire class Borel class measurable function Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence Borel set Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) symmetrizable function Mathematics uniform function Classification of real functions; Baire classification of sets and functions
Content Type	Text
Resource Type	Article
Subject	Mathematics

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