# Sierpiński's hierarchy and locally Lipschitz functions

Fundamenta Mathematicae (1995)

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

## Access Full Article

top## Abstract

top## How to cite

topMorayne, 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

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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.