Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions
Brown, Jack ; Elalaoui-Talibi, Hussain
Colloquium Mathematicae, Tome 79 (1999), p. 277-286 / Harvested from The Polish Digital Mathematics Library
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ℒ denotes the Lebesgue measurable subsets of ℝ and ℒ0 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$ℒ0hasaperfectsubsetQ∈ℒ$ℒ0 which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes ℒ0). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal (s0) 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 ∈ (s0) 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 Gδ sets which can be used to give similar “Marczewski-Burstin-like” characterizations of the collections Bw (sets with the Baire property in the wide sense) and FC (first category sets). It is shown that no collection of Fσ 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 Br (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 U0 (universal null sets). Marczewski-Burstin-like characterizations of the classes of measurable functions are also discussed.

Publié le : 1999-01-01
EUDML-ID : urn:eudml:doc:210762 
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     title = {Marczewski-Burstin-like characterizations of $\sigma$-algebras, ideals, and measurable functions},
     journal = {Colloquium Mathematicae},
     volume = {79},
     year = {1999},
     pages = {277-286},
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Brown, Jack; Elalaoui-Talibi, Hussain. Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions. Colloquium Mathematicae, Tome 79 (1999) pp. 277-286. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv82i2p277bwm/

[000] [1] M. Balcerzak, A. Bartoszewicz, J. Rzepecka and S. Wroński, Marczewski fields and ideals, preprint. | Zbl 1009.28001

[001] [2] S. Baldwin and J. Brown, A simple proof that (s)/(s0) is a complete Boolean algebra, Real Anal. Exchange, to appear. | Zbl 0967.28002

[002] [3] C. Burstin, Eigenschaften messbarer und nicht messbarer Mengen, Sitzungsber. Kaiserlichen Akad. Wiss. Math. Natur. Kl. Abt. IIa 123 (1914), 1525-1551. | Zbl 45.0126.05

[003] [4] C. Kuratowski, La propriété de Baire dans les espaces métriques, Fund. Math. 16 (1930), 390-394. | Zbl 56.0846.03

[004] [5] E. Marczewski (Szpilrajn), Sur une classe de fonctions de M. Sierpiński et la classe correspondante d'ensembles, ibid. 24 (1935), 17-34.

[005] [6] E. Marczewski (Szpilrajn), Sur les ensembles et les fonctions absolument mesurables, C. R. Soc. Sci. Varsovie 30 (1937), 39-68. | Zbl 0017.20302

[006] [7] J. Morgan II, Measurability and the abstract Baire property, Rend. Circ. Mat. Palermo (2) 34 (1985), 234-244. | Zbl 0582.28005

[007] [8] J. Morgan II, Point Set Theory, Marcel Dekker, New York and Basel, 1990.

[008] [9] O. Nikodym, Sur la condition de Baire, Bull. Internat. Acad. Polon. 1929, 591-598.

[009] [10] P. Reardon, Ramsey, Lebesgue, and Marczewski sets and the Baire property, Fund. Math. 149 (1996), 191-203. | Zbl 0846.28002

[010] [11] M. Ruziewicz, Sur une propriété générale des fonctions, Mathematica (Cluj) 9 (1935), 83-85. | Zbl 61.1102.02

[011] [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. | Zbl 61.0228.02

[012] [13] J. Walsh, Marczewski sets, measure and the Baire property, II, Proc. Amer. Math. Soc. 106 (1989), 1027-1030. | Zbl 0671.28002