Wyniki wyszukiwania

1

Strong meager properties for filters

100%

Laflamme C.

Fundamenta Mathematicae

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1994-1995

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tom 146

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nr 3

283-293

EN

We analyze several "strong meager" properties for filters on the natural numbers between the classical Baire property and a filter being $F_σ$. Two such properties have been studied by Talagrand and a few more combinatorial ones are investigated. In particular, we define the notion of a P⁺-filter, a generalization of the traditional concept of P-filter, and prove the existence of a non-meager P⁺-filter. Our motivation lies in understanding the structure of filters generated by complements of members of a maximal almost disjoint family.

2

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

100%

Brown J. B., Elalaoui-Talibi H.

Colloquium Mathematicum

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1999

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tom 82

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nr 2

277-286

EN

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\$ℒ_0$ 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.

3

Ramsey, Lebesgue, and Marczewski sets and the Baire property

63%

Reardon P.

Fundamenta Mathematicae

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1996

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tom 149

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nr 3

191-203

EN

We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented. THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets. THEOREM. In the Ellentuck topology on $[ω]^ω$, $(s)_0$ is a proper subset of the hereditary ideal associated with (s). We construct an example in the Ellentuck topology of a set which is first category and measure 0 but which is not $B_r$-measurable. In addition, several theorems concerning perfect sets in the Ellentuck topology are presented. In particular, it is shown that there exist countable perfect sets in the Ellentuck topology.

Ograniczanie wyników

2 Fundamenta Mathematicae

1 Colloquium Mathematicum

1 Brown J. B.

1 Elalaoui-Talibi H.

1 Laflamme C.

1 Reardon P.

1 1999

1 1996

1 1995

Strong meager properties for filters

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

Ramsey, Lebesgue, and Marczewski sets and the Baire property