Strongly meager sets can be quite big

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2 7 N ov 2 00 1 STRONGLY MEAGER SETS CAN BE QUITE BIG

Lemma 1. 1. I is a σ-ideal, 2. I ⊆ (s)0, 3. I 6 = (s)0 (in ZFC). Notice that such a σ – ideal was defined and investigated in several papers, see for example [4]. Since strongly meager sets and strong measure zero sets are (s)0 it makes sense to ask if they are in I. It is well-known that SN ⊆ I. In fact, if F : 2 −→ 2 is a continuous function and X ∈ SN then F”(X) ∈ SN . The purpose of this pa...

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Strongly Meager Sets Are Not an Ideal

A set X ⊆ R is strongly meager if for every measure zero set H, X + H = R. Let SM denote the collection of strongly meager sets. We show that assuming CH, SM is not an ideal.

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Strongly meager sets and subsets of the plane

Let X ⊆ 2 . Consider the class of all Borel F ⊆ X × 2 with null vertical sections Fx, x ∈ X. We show that if for all such F and all null Z ⊆ X, ⋃ x∈Z Fx is null, then for all such F , ⋃ x∈X Fx 6= 2 . The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P]. A Sierpiński set is an uncountable subset of 2 which meets every null (i.e., measure zero) se...

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We prove the following theorems: (1) Suppose that f : 2ω → 2ω is a continuous function and X is a Sierpiński set. Then (A) for any strongly measure zero set Y , the image f [X + Y ] is an s0-set, (B) f [X] is a perfectly meager set in the transitive sense. (2) Every strongly meager set is completely Ramsey null. This paper is a continuation of earlier works by the authors and by M. Scheepers (s...

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ژورنال

عنوان ژورنال: Israel Journal of Mathematics

سال: 2004

ISSN: 0021-2172,1565-8511

DOI: 10.1007/bf02787551