Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions

Jack Brown; Hussain Elalaoui-Talibi

Colloquium Mathematicae (1999)

  • Volume: 82, Issue: 2, page 277-286
  • ISSN: 0010-1354

Abstract

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ℒ denotes the Lebesgue measurable subsets of ℝ and ℒ 0 denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ$ℒ0 h a s a p e r f e c t s u b s e t Q ∈ ℒ $ ℒ 0 which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes ℒ 0 ). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal ( s 0 ) which we call the “Marczewski null sets”. M ∈ (s) if every perfect set P has a perfect subset Q which is a subset of or misses M. M ∈ ( s 0 ) if every perfect set P has a perfect subset Q which misses M. In this paper, it is shown that there is a collection G of G δ sets which can be used to give similar “Marczewski-Burstin-like” characterizations of the collections B w (sets with the Baire property in the wide sense) and FC (first category sets). It is shown that no collection of F σ sets can be used for this purpose. It is then shown that no collection of Borel sets can be used in a similar way to provide Marczewski-Burstin-like characterizations of B r (sets with the Baire property in the restricted sense) and AFC (always first category sets). The same is true for U (universally measurable sets) and U 0 (universal null sets). Marczewski-Burstin-like characterizations of the classes of measurable functions are also discussed.

How to cite

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Brown, Jack, and Elalaoui-Talibi, Hussain. "Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions." Colloquium Mathematicae 82.2 (1999): 277-286. <http://eudml.org/doc/210762>.

@article{Brown1999,
abstract = {ℒ denotes the Lebesgue measurable subsets of ℝ and $ℒ_0$ denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ$ℒ0$ has a perfect subset Q ∈ ℒ\$ℒ\_0$ which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes $ℒ\_0$). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal $(s^0)$ which we call the “Marczewski null sets”. M ∈ (s) if every perfect set P has a perfect subset Q which is a subset of or misses M. M ∈ $(s^0)$ if every perfect set P has a perfect subset Q which misses M. In this paper, it is shown that there is a collection G of $G\_δ$ sets which can be used to give similar “Marczewski-Burstin-like” characterizations of the collections $B\_w$ (sets with the Baire property in the wide sense) and FC (first category sets). It is shown that no collection of $F\_σ$ sets can be used for this purpose. It is then shown that no collection of Borel sets can be used in a similar way to provide Marczewski-Burstin-like characterizations of $B\_r$ (sets with the Baire property in the restricted sense) and AFC (always first category sets). The same is true for U (universally measurable sets) and $U\_0$ (universal null sets). Marczewski-Burstin-like characterizations of the classes of measurable functions are also discussed.},
author = {Brown, Jack, Elalaoui-Talibi, Hussain},
journal = {Colloquium Mathematicae},
keywords = {Baire property; Marczewski measurable; Lebesgue measurable; Marczewski; -set; Lebesgue measurable subsets; Baire property in the wide sense; Baire property in the restricted sense; classes of measurable functions},
language = {eng},
number = {2},
pages = {277-286},
title = {Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions},
url = {http://eudml.org/doc/210762},
volume = {82},
year = {1999},
}

TY - JOUR
AU - Brown, Jack
AU - Elalaoui-Talibi, Hussain
TI - Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions
JO - Colloquium Mathematicae
PY - 1999
VL - 82
IS - 2
SP - 277
EP - 286
AB - ℒ denotes the Lebesgue measurable subsets of ℝ and $ℒ_0$ denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ$ℒ0$ has a perfect subset Q ∈ ℒ\$ℒ_0$ which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes $ℒ_0$). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal $(s^0)$ which we call the “Marczewski null sets”. M ∈ (s) if every perfect set P has a perfect subset Q which is a subset of or misses M. M ∈ $(s^0)$ if every perfect set P has a perfect subset Q which misses M. In this paper, it is shown that there is a collection G of $G_δ$ sets which can be used to give similar “Marczewski-Burstin-like” characterizations of the collections $B_w$ (sets with the Baire property in the wide sense) and FC (first category sets). It is shown that no collection of $F_σ$ sets can be used for this purpose. It is then shown that no collection of Borel sets can be used in a similar way to provide Marczewski-Burstin-like characterizations of $B_r$ (sets with the Baire property in the restricted sense) and AFC (always first category sets). The same is true for U (universally measurable sets) and $U_0$ (universal null sets). Marczewski-Burstin-like characterizations of the classes of measurable functions are also discussed.
LA - eng
KW - Baire property; Marczewski measurable; Lebesgue measurable; Marczewski; -set; Lebesgue measurable subsets; Baire property in the wide sense; Baire property in the restricted sense; classes of measurable functions
UR - http://eudml.org/doc/210762
ER -

References

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  2. [2] S. Baldwin and J. Brown, A simple proof that ( s ) / ( s 0 ) is a complete Boolean algebra, Real Anal. Exchange, to appear. Zbl0967.28002
  3. [3] C. Burstin, Eigenschaften messbarer und nicht messbarer Mengen, Sitzungsber. Kaiserlichen Akad. Wiss. Math. Natur. Kl. Abt. IIa 123 (1914), 1525-1551. Zbl45.0126.05
  4. [4] C. Kuratowski, La propriété de Baire dans les espaces métriques, Fund. Math. 16 (1930), 390-394. Zbl56.0846.03
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  8. [8] J. Morgan II, Point Set Theory, Marcel Dekker, New York and Basel, 1990. 
  9. [9] O. Nikodym, Sur la condition de Baire, Bull. Internat. Acad. Polon. 1929, 591-598. 
  10. [10] P. Reardon, Ramsey, Lebesgue, and Marczewski sets and the Baire property, Fund. Math. 149 (1996), 191-203. Zbl0846.28002
  11. [11] M. Ruziewicz, Sur une propriété générale des fonctions, Mathematica (Cluj) 9 (1935), 83-85. Zbl61.1102.02
  12. [12] W. Sierpiński, Sur un problème de M. Ruziewicz concernant les superpositions de fonctions jouissant de la propriété de Baire, Fund. Math. 24 (1935), 12-16. Zbl61.0228.02
  13. [13] J. Walsh, Marczewski sets, measure and the Baire property, II, Proc. Amer. Math. Soc. 106 (1989), 1027-1030. Zbl0671.28002

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