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Strong Measure Zero and Strongly Meager Sets

1993
*
Proceedings of the American Mathematical Society
*

We consider conjectures made by Prikry

doi:10.2307/2160341
fatcat:icbibosc3vdq5mvbisp53etgty
*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. ...##
###
Strongly meager and strong measure zero sets

2002
*
Archive for Mathematical Logic
*

In this paper we present two consistency results concerning the existence of large

doi:10.1007/s001530000068
fatcat:vkyjbo3ryfae7lwe2mcmoh26w4
*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*...##
###
Strongly meager and strong measure zero sets
[article]

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). ...

##
###
Strong measure zero and strongly meager sets

1993
*
Proceedings of the American Mathematical Society
*

We consider conjectures made by Prikry

doi:10.1090/s0002-9939-1993-1139474-6
fatcat:khvhdytvvvhmzedpmlqj6wlpha
*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. ...##
###
Cohen real or random real: effect on strong measure zero sets and strongly meager sets
[article]

2020
*
arXiv
*
pre-print

We show that the

arXiv:2005.07912v2
fatcat:exrmgkuftbf6plofanknzd2xqi
*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. ...##
###
Products of special sets of real numbers
[article]

2007
*
arXiv
*
pre-print

The product of a

arXiv:math/0307226v4
fatcat:o3fr6fhlybcmbil3ko4774gfqu
*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. ...##
###
Products of Special Sets of Real Numbers

2005
*
Real Analysis Exchange
*

The product of a

doi:10.14321/realanalexch.30.2.0819
fatcat:2ztreazddzgc5f2aqvcjdtws6e
*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. ...##
###
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]

2001
*
arXiv
*
pre-print

We show that the Dual Borel Conjecture implies that d> _1

arXiv:math/0107190v1
fatcat:eiu2jnrmbjhc5dar2d3gkop2vq
*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. ...##
###
On Sierpinski Sets

1990
*
Proceedings of the American Mathematical Society
*

We prove that it is consistent with ZFC that every Sierpinski

doi:10.2307/2048302
fatcat:hckdcacn3bf7nem4t7fm23tyme
*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. ...##
###
On Sierpiński sets

1990
*
Proceedings of the American Mathematical Society
*

We prove that it is consistent with ZFC that every Sierpinski

doi:10.1090/s0002-9939-1990-0991689-0
fatcat:mij3nhosircvxfk45465gsh6m4
*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. ...
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