Sierpiński's hierarchy and locally Lipschitz functions

Michał Morayne

Fundamenta Mathematicae (1995)

  • Volume: 147, Issue: 1, page 73-82
  • ISSN: 0016-2736

Abstract

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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 < ω 1 then f ○ g ∈ B α ( Z ) for every g ∈ B α ( Z ) ∩ Z I if and only if f is continuous on I, where B α ( Z ) stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes S α ( 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 S 1 ( Z ) ). This theorem solves the problem raised by the work of Lindenbaum ([L] and [L, Corr.]) concerning the class of functions not leading outside S α ( Z ) by outer superpositions.

How to cite

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Morayne, Michał. "Sierpiński's hierarchy and locally Lipschitz functions." Fundamenta Mathematicae 147.1 (1995): 73-82. <http://eudml.org/doc/212075>.

@article{Morayne1995,
abstract = {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 $α < ω_1$ then f ○ g ∈ $B_α(Z)$ for every $g ∈ B_α(Z) ∩^ZI$ if and only if f is continuous on I, where $B_α(Z)$ stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes $S_α(Z) (α > 0)$ 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 $S_1(Z)$). This theorem solves the problem raised by the work of Lindenbaum ([L] and [L, Corr.]) concerning the class of functions not leading outside $S_α(Z)$ by outer superpositions.},
author = {Morayne, Michał},
journal = {Fundamenta Mathematicae},
keywords = {locally Lipschitz functions; Sierpiński classification of Borel measurable functions},
language = {eng},
number = {1},
pages = {73-82},
title = {Sierpiński's hierarchy and locally Lipschitz functions},
url = {http://eudml.org/doc/212075},
volume = {147},
year = {1995},
}

TY - JOUR
AU - Morayne, Michał
TI - Sierpiński's hierarchy and locally Lipschitz functions
JO - Fundamenta Mathematicae
PY - 1995
VL - 147
IS - 1
SP - 73
EP - 82
AB - 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 $α < ω_1$ then f ○ g ∈ $B_α(Z)$ for every $g ∈ B_α(Z) ∩^ZI$ if and only if f is continuous on I, where $B_α(Z)$ stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes $S_α(Z) (α > 0)$ 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 $S_1(Z)$). This theorem solves the problem raised by the work of Lindenbaum ([L] and [L, Corr.]) concerning the class of functions not leading outside $S_α(Z)$ by outer superpositions.
LA - eng
KW - locally Lipschitz functions; Sierpiński classification of Borel measurable functions
UR - http://eudml.org/doc/212075
ER -

References

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  1. [CM1] J. Cichoń and M. Morayne, Universal functions and generalized classes of functions, Proc. Amer. Math. Soc. 102 (1988), 83-89. Zbl0646.26009
  2. [CM2] J. Cichoń and M. Morayne, An abstract version of Sierpiński's theorem and the algebra generated by A and CA functions, Fund. Math. 142 (1993), 263-268. Zbl0824.54009
  3. [H] F. Hausdorff, Set Theory, Chelsea, New York, 1962. 
  4. [Ke] S. Kempisty, Sur les séries itérées des fonctions continues, Fund. Math. 2 (1921), 64-73. Zbl48.0276.04
  5. [Ku] K. Kuratowski, Topology I, Academic Press, New York and London, 1966. 
  6. [L] A. Lindenbaum, Sur les superpositions de fonctions représentables analytiquement, Fund. Math. 23 (1934), 15-37; Corrections, ibid., 304. Zbl60.0195.02
  7. [Mau] R. D. Mauldin, Baire functions, Borel sets, and ordinary function systems, Adv. in Math. 12 (1974), 418-450. Zbl0278.26005
  8. [Maz] S. Mazurkiewicz, Sur les fonctions de classe 1, Fund. Math. 2 (1921), 28-36. 
  9. [Mor] M. Morayne, Algebras of Borel measurable functions, ibid. 141 (1992), 229-242. Zbl0812.26004
  10. [S1] W. Sierpiński, Sur les fonctions développables en séries absolument convergentes de fonctions continues, ibid. 2 (1921), 15-27. Zbl48.0276.01
  11. [S2] W. Sierpiński, Démonstration d'un théorème sur les fonctions de première classe, ibid. 2 (1921), 37-40. Zbl48.0276.03

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