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Strong Measure Zero and Strongly Meager Sets

Timothy J. Carlson
1993 Proceedings of the American Mathematical Society  
We consider conjectures made by Prikry and Galvin concerning strong measure zero and strongly meager sets of real numbers.  ...  Galvin, Mycielski, and Solovay proved a conjecture of Prikry by showing that a set is of strong measure zero iff it is contained in a translate of every comeager set.  ...  For suppose that Xi is strongly meager for i e k and let X be the union of the Xi. Now suppose that A is a set of full measure and get B from Problem 5.1.  ... 
doi:10.2307/2160341 fatcat:icbibosc3vdq5mvbisp53etgty

Strongly meager and strong measure zero sets

Tomek Bartoszyński, Saharon Shelah
2002 Archive for Mathematical Logic  
In this paper we present two consistency results concerning the existence of large strong measure zero and strongly meager sets.  ...  We say that X is a strong measure zero set if for every F ∈ M, X +F = 2 ω . Let SM denote the collection of strongly meager sets and let SN denote the collection of strong measure zero sets.  ...  modified:1999-08-10 revision:1999-07-22 modified:1999-08-10 STRONGLY MEAGER AND STRONG MEASURE ZERO SETS  ... 
doi:10.1007/s001530000068 fatcat:vkyjbo3ryfae7lwe2mcmoh26w4

Strongly meager and strong measure zero sets [article]

Tomek Bartoszynski, Saharon Shelah
1999 arXiv   pre-print
Strongly meager sets form an ideal with the same additivity as the ideal of meager sets. 2. There exists a strong measure zero set of size > d (dominating number).  ...  We say that X is a strong measure zero set if for every F ∈ M, X + F = 2 ω . Let SM denote the collection of strongly meager sets and let SN denote the collection of strong measure zero sets.  ...  To witness that a set is not strongly meager we need a measure zero set. The following theorem is crucial. Theorem 2.1 (Lorentz).  ... 
arXiv:math/9907137v1 fatcat:pjptcfppu5f7xedoeseied7bhu

Strong measure zero and strongly meager sets

Timothy J. Carlson
1993 Proceedings of the American Mathematical Society  
We consider conjectures made by Prikry and Galvin concerning strong measure zero and strongly meager sets of real numbers.  ...  Galvin, Mycielski, and Solovay proved a conjecture of Prikry by showing that a set is of strong measure zero iff it is contained in a translate of every comeager set.  ...  For suppose that Xi is strongly meager for i e k and let X be the union of the Xi. Now suppose that A is a set of full measure and get B from Problem 5.1.  ... 
doi:10.1090/s0002-9939-1993-1139474-6 fatcat:khvhdytvvvhmzedpmlqj6wlpha

Cohen real or random real: effect on strong measure zero sets and strongly meager sets [article]

Miguel A. Cardona
2020 arXiv   pre-print
We show that the set of the ground-model reals has strong measure zero (is strongly meager) after adding a single Cohen real (random real).  ...  As consequence we prove that the set of the ground-model reals has strong measure zero after adding a single Hechler real.  ...  and Geometry, TU Wien.  ... 
arXiv:2005.07912v2 fatcat:exrmgkuftbf6plofanknzd2xqi

Products of special sets of real numbers [article]

Boaz Tsaban, Tomasz Weiss
2007 arXiv   pre-print
The product of a meager/null-additive set and a strong measure zero/strongly meager set in the Cantor space has strong measure zero/is strongly meager, respectively. 2.  ...  These results extend results of Scheepers and Miller, respectively.  ...  We thank Marcin Kysiak for his comment following Theorem 2.3, and Tomek Bartoszyński for the permission to include here his proof of Theorem A.2.  ... 
arXiv:math/0307226v4 fatcat:o3fr6fhlybcmbil3ko4774gfqu

Products of Special Sets of Real Numbers

Boaz Tsaban, Tomasz Weiss
2005 Real Analysis Exchange  
The product of a meager/null-additive set and a strong measure zero/strongly meager set in the Cantor space has strong measure zero/is strongly meager, respectively. 2.  ...  These results extend results of Scheepers and Miller, respectively.  ...  We thank Marcin Kysiak for his comment following Theorem 2.3, and Tomek Bartoszyński for the permission to include here his proof of Theorem A.2.  ... 
doi:10.14321/realanalexch.30.2.0819 fatcat:2ztreazddzgc5f2aqvcjdtws6e

Page 9546 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
X is strongly measure zero if for every meager set M ©2°,X+M £2”. Similarly, Y is strongly meager if for every measure zero set N C2”, ¥ + N #2”.  ...  X is strongly measure zero (XY € SN) if for every meager set MC 2”, X + M #2”. Similarly, X is strongly meager (XY € SM) if for every measure zero set N C2, ¥ +N 42°.  ... 

Page 2331 of Mathematical Reviews Vol. , Issue 2002D [page]

2002 Mathematical Reviews  
Using Borel’s definition it is easy to see that the strong measure zero sets form a a-ideal, i.e., the union of countably many strong measure zero sets has strong measure zero.  ...  Borel introduced the notion of strong measure zero sets in 1919.  ... 

Page 8593 of Mathematical Reviews Vol. , Issue 2001M [page]

2001 Mathematical Reviews  
It is also shown that Martin’s Axiom implies that there exist a strongly meager set XY and a universally measure zero set Y such that ¥ + Y contains an uncountable closed set. Arnold W.  ...  In this paper it is shown that Martin’s Axiom implies that there exist a universally meager set X and a strong measure zero set Y such that X¥ + Y contains an uncountable closed set.  ... 

Page 809 of Mathematical Reviews Vol. , Issue 2003B [page]

2003 Mathematical Reviews  
Let SM c P(2”) be the family of all strongly meager sets and SN c P(2”) the family of all strongly measure zero sets.  ...  Recall that a subset X C 2° is called strongly measure zero [strongly meager] if for every meager [measure zero] set Y C 2° we have X + Y #2”, where addition is componentwise modulo 2.  ... 

Page 619 of Mathematical Reviews Vol. , Issue 94b [page]

1994 Mathematical Reviews  
Strong measure zero and strongly meager sets. Proc. Amer. Math. Soc. 118 (1993), no. 2, 577-586.  ...  The author considers several conjectures made by Prikry and Galvin about the strong measure zero sets and the strong meager sets.  ... 

Remarks on small sets of reals [article]

Tomek Bartoszynski
2001 arXiv   pre-print
We show that the Dual Borel Conjecture implies that d> _1 and find some topological characterizations of perfectly meager and universally meager sets.  ...  Let Borel Conjecture be the statement that there are no uncountable strong measure zero sets, and the Dual Borel Conjecture that there are no uncountable strongly meager sets.  ...  X has strong measure zero if X + F = 2 ω for all F ∈ M, 2. X is strongly meager if X + F = 2 ω for all F ∈ N , 3. X is meager additive if X + F ∈ M for all F ∈ M, 4.  ... 
arXiv:math/0107190v1 fatcat:eiu2jnrmbjhc5dar2d3gkop2vq

On Sierpinski Sets

Tomek Bartoszynski, Haim Judah
1990 Proceedings of the American Mathematical Society  
We prove that it is consistent with ZFC that every Sierpinski set is strongly meager. It is also proved that under CH every Sierpinski set is a union of two strongly meager sets.  ...  In other words every Lusin set has strong measure zero (see [M] for this and many related results).  ...  A set S ç R is called a Sierpinski set if S is uncountable and Sí)H is countable for every measure zero set H ç R.  ... 
doi:10.2307/2048302 fatcat:hckdcacn3bf7nem4t7fm23tyme

On Sierpiński sets

Tomek Bartoszy{ński, Haim Judah
1990 Proceedings of the American Mathematical Society  
We prove that it is consistent with ZFC that every Sierpinski set is strongly meager. It is also proved that under CH every Sierpinski set is a union of two strongly meager sets.  ...  In other words every Lusin set has strong measure zero (see [M] for this and many related results).  ...  A set S ç R is called a Sierpinski set if S is uncountable and Sí)H is countable for every measure zero set H ç R.  ... 
doi:10.1090/s0002-9939-1990-0991689-0 fatcat:mij3nhosircvxfk45465gsh6m4
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