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Strong measure zero and meager-additive sets through the prism of fractal measures. (English) Zbl 07088828
Summary: 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 \(f\colon 2^{\omega}\to 2^{\omega}\) is continuous and \(X\) is meager-additive, then so is \(f(X)\).

MSC:
03E05 Other combinatorial set theory
03E20 Other classical set theory (including functions, relations, and set algebra)
28A78 Hausdorff and packing measures
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[1] Bartoszyński T.; Judah H., Set Theory, On the structure of the real line. A K Peters, Wellesley, 1995
[2] Bartoszyński T.; Shelah S., Closed measure zero sets, Ann. Pure Appl. Logic 58 (1992), no. 2, 93-110
[3] Besicovitch A. S., Concentrated and rarified sets of points, Acta Math. 62 (1933), no. 1, 289-300
[4] Besicovitch A. S., Correction, Acta Math. 62 (1933), no. 1, 317-318
[5] Borel E., Sur la classification des ensembles de mesure nulle, Bull. Soc. Math. France 47 (1919), 97-125 (French)
[6] Carlson T. J., Strong measure zero and strongly meager sets, Proc. Amer. Math. Soc. 118 (1993), no. 2, 577-586
[7] Corazza P., The generalized Borel conjecture and strongly proper orders, Trans. Amer. Math. Soc. 316 (1989), no. 1, 115-140
[8] Federer H., Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, 153, Springer, New York, 1969
[9] Fremlin D. H., Measure Theory. Vol. 5, Set-theoretic Measure Theory, Part I, Torres Fremlin, Colchester, 2015
[10] Galvin F.; Miller A. W., \(\gamma \), -sets and other singular sets of real numbers, Topology Appl. 17 (1984), no. 2, 145-155
[11] Galvin F.; Mycielski J.; Solovay R. M., Strong measure zero sets, Abstract 79T-E25, Not. Am. Math. Soc. 26 (1979), A-280
[12] Galvin F.; Mycielski J.; Solovay R. M., Strong measure zero and infinite games, Arch. Math. Logic 56 (2017), no. 7-8, 725-732
[13] Gerlits J.; Nagy Z., Some properties of \(C(X)\), I, Topology Appl. 14 (1982), no. 2, 151-161
[14] Gödel K., The consistency of the axiom of choice and of the generalized continuum-hypothesis, Proc. Natl. Acad. Sci. USA 24 (1938), no. 12, 556-557
[15] Gödel K., The Consistency of the Continuum Hypothesis, Annals of Mathematics Studies, 3, Princeton University Press, Princeton, 1940
[16] Howroyd J. D., On the Theory of Hausdorff Measures in Metric Spaces, Ph.D. Thesis, University College, London, 1994
[17] Howroyd J. D., On Hausdorff and packing dimension of product spaces, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 4, 715-727
[18] Hrušák M.; Wohofsky W.; Zindulka O., Strong measure zero in separable metric spaces and Polish groups, Arch. Math. Logic 55 (2016), no. 1-2, 105-131
[19] Hrušák M.; Zapletal J., Strong measure zero sets in Polish groups, Illinois J. Math. 60 (2016), no. 3-4, 751-760
[20] Kelly J. D., A method for constructing measures appropriate for the study of Cartesian products, Proc. London Math. Soc. (3) 26 (1973), 521-546
[21] Kysiak M., On Erdős-Sierpiński Duality between Lebesgue Measure and Baire Category, Master’s Thesis, Uniwersytet Warszawski, Warszawa, 2000 (Polish)
[22] Laver R., On the consistency of Borel’s conjecture, Acta Math. 137 (1976), no. 3-4, 151-169
[23] Munroe M. E., Introduction to Measure and Integration, Addison-Wesley Publishing Company, Cambridge, 1953
[24] Nowik A.; Scheepers M.; Weiss T., The algebraic sum of sets of real numbers with strong measure zero sets, J. Symbolic Logic 63 (1998), no. 1, 301-324
[25] Nowik A.; Weiss T., On the Ramseyan properties of some special subsets of \(2^\omega\) and their algebraic sums, J. Symbolic Logic 67 (2002), no. 2, 547-556
[26] Pawlikowski J., A characterization of strong measure zero sets, Israel J. Math. 93 (1996), 171-183
[27] Rogers C. A., Hausdorff Measures, Cambridge University Press, London, 1970
[28] Scheepers M., Finite powers of strong measure zero sets, J. Symbolic Logic 64 (1999), no. 3, 1295-1306
[29] Shelah S., Every null-additive set is meager-additive, Israel J. Math. 89 (1995), no. 1-3, 357-376
[30] Sierpiński W., Sur un ensemble non denombrable, dont toute image continue est de mesure nulle, Fundamenta Mathematicae 11 (1928), no. 1, 302-304 (French)
[31] Tsaban B.; Weiss T., Products of special sets of real numbers, Real Anal. Exchange 30 (2004/05), no. 2, 819-835
[32] Weiss T., On meager additive and null additive sets in the Cantor space \(2^\omega\) and in \(\Bbb R\), Bull. Pol. Acad. Sci. Math. 57 (2009), no. 2, 91-99
[33] Weiss T., Addendum to “On meager additive and null additive sets in the Cantor space \(2^\omega\) and in \(\Bbb R\)” (Bull. Polish Acad. Sci. Math. 57 (2009), 91-99), Bull. Pol. Acad. Sci. Math. 62 (2014), no. 1, 1-9
[34] Weiss T., Properties of the intersection ideal \(\mathcal M\cap \mathcal N\) revisited, Bull. Pol. Acad. Sci. Math. 65 (2017), no. 2, 107-111
[35] Weiss T.; Tsaban B., Topological diagonalizations and Hausdorff dimension, Note Mat. 22 (2003/04), no. 2, 83-92
[36] Wohofsky W., Special Sets of Real Numbers and Variants of the Borel Conjecture, Ph.D. Thesis, Technische Universität Wien, Wien, 2013
[37] Zakrzewski P., Universally meager sets, Proc. Amer. Math. Soc. 129 (2001), no. 6, 1793-1798
[38] Zakrzewski P., Universally meager sets. II, Topology Appl. 155 (2008), no. 13, 1445-1449
[39] Zindulka O., Universal measure zero, large Hausdorff dimension, and nearly Lipschitz maps, Fund. Math. 218 (2012), no. 2, 95-119
[40] Zindulka O., Packing measures and dimensions on Cartesian products, Publ. Mat. 57 (2013), no. 2, 393-420
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