Strong measure zero and meager-additive sets through the prism of fractal measures

Ondřej Zindulka

Commentationes Mathematicae Universitatis Carolinae (2019)

  • Volume: 60, Issue: 1, page 131-155
  • ISSN: 0010-2628

Abstract

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We develop a theory of sharp measure zero sets that parallels Borel’s strong measure zero, and prove a theorem analogous to Galvin–Mycielski–Solovay theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of 2 ω is meager-additive if and only if it is ℰ -additive; if f : 2 ω → 2 ω is continuous and X is meager-additive, then so is f ( X ) .

How to cite

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Zindulka, Ondřej. "Strong measure zero and meager-additive sets through the prism of fractal measures." Commentationes Mathematicae Universitatis Carolinae 60.1 (2019): 131-155. <http://eudml.org/doc/294665>.

@article{Zindulka2019,
abstract = {We develop a theory of sharp measure zero sets that parallels Borel’s strong measure zero, and prove a theorem analogous to Galvin–Mycielski–Solovay theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of $2^\{\omega \}$ is meager-additive if and only if it is $\mathcal \{E\}$-additive; if $f\colon 2^\{\omega \}\rightarrow 2^\{\omega \}$ is continuous and $X$ is meager-additive, then so is $f(X)$.},
author = {Zindulka, Ondřej},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {meager-additive; $\mathcal \{E\}$-additive; strong measure zero; sharp measure zero; Hausdorff dimension; Hausdorff measure},
language = {eng},
number = {1},
pages = {131-155},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Strong measure zero and meager-additive sets through the prism of fractal measures},
url = {http://eudml.org/doc/294665},
volume = {60},
year = {2019},
}

TY - JOUR
AU - Zindulka, Ondřej
TI - Strong measure zero and meager-additive sets through the prism of fractal measures
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 1
SP - 131
EP - 155
AB - We develop a theory of sharp measure zero sets that parallels Borel’s strong measure zero, and prove a theorem analogous to Galvin–Mycielski–Solovay theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of $2^{\omega }$ is meager-additive if and only if it is $\mathcal {E}$-additive; if $f\colon 2^{\omega }\rightarrow 2^{\omega }$ is continuous and $X$ is meager-additive, then so is $f(X)$.
LA - eng
KW - meager-additive; $\mathcal {E}$-additive; strong measure zero; sharp measure zero; Hausdorff dimension; Hausdorff measure
UR - http://eudml.org/doc/294665
ER -

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