Sierpiński's hierarchy and locally Lipschitz functions
Morayne, Michał
Fundamenta Mathematicae, Tome 146 (1995), p. 73-82 / Harvested from The Polish Digital Mathematics Library
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Let Z be an uncountable Polish space. It is a classical result that if I ⊆ ℝ is any interval (proper or not), f: I → ℝ and α<ω1 then f ○ g ∈ Bα(Z) for every g∈Bα(Z)∩ZI if and only if f is continuous on I, where Bα(Z) stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes Sα(Z)(α>0) in Sierpiński’s classification of Borel measurable functions the analogous result holds where the condition that f is continuous is replaced by the condition that f is locally Lipschitz on I (thus it holds for the class of differences of semicontinuous functions, which is the class S1(Z)). This theorem solves the problem raised by the work of Lindenbaum ([L] and [L, Corr.]) concerning the class of functions not leading outside Sα(Z) by outer superpositions.

Publié le : 1995-01-01
EUDML-ID : urn:eudml:doc:212075 
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     title = {Sierpi\'nski's hierarchy and locally Lipschitz functions},
     journal = {Fundamenta Mathematicae},
     volume = {146},
     year = {1995},
     pages = {73-82},
     zbl = {0833.26006},
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Morayne, Michał. Sierpiński's hierarchy and locally Lipschitz functions. Fundamenta Mathematicae, Tome 146 (1995) pp. 73-82. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv147i1p73bwm/

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