Continuous-, derivative-, and differentiable-restrictions of measurable functions

Jack Brown

Fundamenta Mathematicae (1992)

  • Volume: 141, Issue: 1, page 85-95
  • ISSN: 0016-2736

Abstract

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We review the known facts and establish some new results concerning continuous-restrictions, derivative-restrictions, and differentiable-restrictions of Lebesgue measurable, universally measurable, and Marczewski measurable functions, as well as functions which have the Baire properties in the wide and restricted senses. We also discuss some known examples and present a number of new examples to show that the theorems are sharp.

How to cite

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Brown, Jack. "Continuous-, derivative-, and differentiable-restrictions of measurable functions." Fundamenta Mathematicae 141.1 (1992): 85-95. <http://eudml.org/doc/211953>.

@article{Brown1992,
abstract = {We review the known facts and establish some new results concerning continuous-restrictions, derivative-restrictions, and differentiable-restrictions of Lebesgue measurable, universally measurable, and Marczewski measurable functions, as well as functions which have the Baire properties in the wide and restricted senses. We also discuss some known examples and present a number of new examples to show that the theorems are sharp.},
author = {Brown, Jack},
journal = {Fundamenta Mathematicae},
keywords = {Lebesgue measurability; Marczewski measurability; universally measurable sets; continuous-restrictions; derivative-restrictions; Baire properties},
language = {eng},
number = {1},
pages = {85-95},
title = {Continuous-, derivative-, and differentiable-restrictions of measurable functions},
url = {http://eudml.org/doc/211953},
volume = {141},
year = {1992},
}

TY - JOUR
AU - Brown, Jack
TI - Continuous-, derivative-, and differentiable-restrictions of measurable functions
JO - Fundamenta Mathematicae
PY - 1992
VL - 141
IS - 1
SP - 85
EP - 95
AB - We review the known facts and establish some new results concerning continuous-restrictions, derivative-restrictions, and differentiable-restrictions of Lebesgue measurable, universally measurable, and Marczewski measurable functions, as well as functions which have the Baire properties in the wide and restricted senses. We also discuss some known examples and present a number of new examples to show that the theorems are sharp.
LA - eng
KW - Lebesgue measurability; Marczewski measurability; universally measurable sets; continuous-restrictions; derivative-restrictions; Baire properties
UR - http://eudml.org/doc/211953
ER -

References

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  1. [1] S. Agronsky, A. M. Bruckner, M. Laczkovich and D. Preiss, Convexity conditions and intersections with smooth functions, Trans. Amer. Math. Soc. 289 (1985), 659-677. Zbl0601.26007
  2. [2] H. Blumberg, New properties of all real functions, ibid. 24 (1922), 113-128. 
  3. [3] J. B. Brown, Differentiable restrictions of real functions, Proc. Amer. Math. Soc. 108 (1990), 391-398. Zbl0692.26002
  4. [4] J. B. Brown and K. Prikry, Variations on Lusin's theorem, Trans. Amer. Math. Soc. 302 (1987), 77-86. Zbl0619.28005
  5. [5] A. M. Bruckner, J. G. Ceder and M. L. Weiss, On the differentiability structure of real functions, ibid. 142 (1969), 1-13. Zbl0182.38301
  6. [6] J. Ceder, Some examples on continuous restrictions, Real Anal. Exchange 7 (1981/ 82), 155-162. Zbl0533.26002
  7. [7] F. Filipczak, Sur les fonctions continues relativement monotones, Fund. Math. 58 (1966), 75-87. Zbl0185.12204
  8. [8] V. Jarník, Sur les nombres dérivés approximatifs, ibid. 22 (1934), 4-16. 
  9. [9] C. Kuratowski, La propriété de Baire dans les espaces métriques, ibid. 16 (1930), 390-394. Zbl56.0846.03
  10. [10] K. Kuratowski and A. Mostowski, Set Theory with an Introduction to Descriptive Set Theory, North-Holland, Amsterdam 1976. Zbl0337.02034
  11. [11] M. Laczkovich, Differentiable restrictions of continuous functions, Acta Math. Hungar. 44 (1984), 355-360. Zbl0558.26005
  12. [12] N. Lusin, Sur les propriétés des fonctions mesurables, C. R. Acad. Sci. Paris 154 (1912), 1688-1690. Zbl43.0484.04
  13. [13] N. Lusin, Sur la recherche des fonctions primitives, ibid. 162 (1916), 975-978. Zbl46.0390.02
  14. [14] E. Marczewski (Szpilrajn), Sur une classe de fonctions de M. Sierpiński et la classe correspondante d'ensembles, Fund. Math. 24 (1935), 17-34. Zbl61.0229.01

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