Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl
Preferencje help
Polish version English version Język
Widoczny [Schowaj] Abstrakt
Liczba wyników

Znaleziono wyników: 5

Liczba wyników na stronie
first rewind previous Strona / 1 next fast forward last

Wyniki wyszukiwania

help Sortuj według:

help Ogranicz wyniki do:
first rewind previous Strona / 1 next fast forward last
1
Content available remote

Measurable envelopes, Hausdorff measures and Sierpiński sets

100%
Elekes M.
Colloquium Mathematicum
|
2003
|
tom 98
|
nr 2
155-162
EN
We show that the existence of measurable envelopes of all subsets of ℝⁿ with respect to the d-dimensional Hausdorff measure (0 < d < n) is independent of ZFC. We also investigate the consistency of the existence of $ℋ ^{d}$-measurable Sierpiński sets.
2
Content available remote

Covering locally compact groups by less than $2^{ω}$ many translates of a compact nullset

64%
Elekes M., Tóth Á.
Fundamenta Mathematicae
|
2007
|
tom 193
|
nr 3
243-257
EN
Gruenhage asked if it was possible to cover the real line by less than continuum many translates of a compact nullset. Under the Continuum Hypothesis the answer is obviously negative. Elekes and Stepr# mans gave an affirmative answer by showing that if $C_{EK}$ is the well known compact nullset considered first by Erdős and Kakutani then ℝ can be covered by cof(𝓝) many translates of $C_{EK}$. As this set has no analogue in more general groups, it was asked by Elekes and Stepr# mans whether such a result holds for uncountable locally compact Polish groups. In this paper we give an affirmative answer in the abelian case. More precisely, we show that if G is a nondiscrete locally compact abelian group in which every open subgroup is of index at most cof(𝓝) then there exists a compact set C of Haar measure zero such that G can be covered by cof(𝓝) many translates of C. This result, which is optimal in a sense, covers the cases of uncountable compact abelian groups and of nondiscrete separable locally compact abelian groups. We use Pontryagin's duality theory to reduce the problem to three special cases; the circle group, countable products of finite discrete abelian groups, and the groups of p-adic integers, and then we solve the problem on these three groups separately. In addition, using representation theory, we reduce the nonabelian case to the classes of Lie groups and profinite groups, and we also settle the problem for Lie groups. (M. Abért recently gave an affirmative answer for profinite groups, so the nonabelian case is also complete.)
3
Content available remote

Less than $2^{ω}$ many translates of a compact nullset may cover the real line

64%
Elekes M., Steprāns J.
Fundamenta Mathematicae
|
2004
|
tom 181
|
nr 1
89-96
EN
We answer a question of Darji and Keleti by proving that there exists a compact set C₀ ⊂ ℝ of measure zero such that for every perfect set P ⊂ ℝ there exists x ∈ ℝ such that (C₀+x) ∩ P is uncountable. Using this C₀ we answer a question of Gruenhage by showing that it is consistent with ZFC (as it follows e.g. from $cof(𝓝) < 2^{ω}$) that less than $2^{ω}$ many translates of a compact set of measure zero can cover ℝ.
4
Content available remote

Can we assign the Borel hulls in a monotone way?

64%
Elekes M., Máthé A.
Fundamenta Mathematicae
|
2009
|
tom 205
|
nr 2
105-115
EN
A hull of A ⊆ [0,1] is a set H containing A such that λ*(H) = λ*(A). We investigate all four versions of the following problem. Does there exist a monotone (with respect to inclusion) map that assigns a Borel/$G_{δ}$ hull to every negligible/measurable subset of [0,1]? Three versions turn out to be independent of ZFC, while in the fourth case we only prove that the nonexistence of a monotone $G_{δ}$ hull operation for all measurable sets is consistent. It remains open whether existence here is also consistent. We also answer the question of Z. Gyenes and D. Pálvölgyi whether monotone hulls can be defined for every chain of measurable sets. Moreover, we comment on the problem of hulls of all subsets of [0,1].
5
Content available remote

On splitting infinite-fold covers

51%
Elekes M., Mátrai T., Soukup L.
Fundamenta Mathematicae
|
2011
|
tom 212
|
nr 2
95-127
EN
Let X be a set, κ be a cardinal number and let ℋ be a family of subsets of X which covers each x ∈ X at least κ-fold. What assumptions can ensure that ℋ can be decomposed into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover ℋ: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of ℝⁿ by polyhedra and by arbitrary convex sets. We focus on problems with κ infinite. Besides numerous positive and negative results, many questions turn out to be independent of the usual axioms of set theory.
first rewind previous Strona / 1 next fast forward last
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.