A fine correlation between Baire and Borel functional hierarchiesстатья
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Аннотация:There are two widely known functional hierarchies on a topological space (T,G) . The transfinite chain A(T)⊂…⊂LimαA(T)⊂…⊂Limω1A(T) , where A(T) is an initial family of functions on T and LimαA(T) consists of all pointwise limits of sequences of functions from preceding classes, is called the Baire convergence hierarchy. The transfinite chain M(T,B0)⊂…⊂M(T,Bα)⊂…⊂M(T,B(T,G)) , where B0⊂…Bα⊂…⊂B(T,G) is some hierarchy in the σ-algebra B(T,G) of Borel sets and M(T,Bα) is the family of all Bα -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,G) and Young–Hausdorff ensembles B0≡G , B1≡Fσ , B2≡Gδσ , B3≡Fσδσ , …, which is valid only for perfectly normal spaces. The second one is the Banach correlation with the initial family M(T,Fσ) , which is valid only for perfect spaces.
For an arbitrary topological space (T,G) there is the general correlation with the initial family M(T,Gλ) , B0≡G , B1≡ Gλ , B2≡Gλλ , where Eλ≡{⋃(En∩(T∖Hn)∣n∈ω)∣En,Hn∈E} .
In this paper we establish the fine Baire–Borel correlation, i. e., we find the initial family of uniform functions strictly intermediate between Cb(T,G) and Mb(T,Gλ) .