Ramsey, Lebesgue, and Marczewski sets and the Baire property

Patrick Reardon

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 3, page 191-203
  • ISSN: 0016-2736

Abstract

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We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented.  THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets.  THEOREM. In the Ellentuck topology on [ ω ] ω , ( s ) 0 is a proper subset of the hereditary ideal associated with (s).  We construct an example in the Ellentuck topology of a set which is first category and measure 0 but which is not B r -measurable. In addition, several theorems concerning perfect sets in the Ellentuck topology are presented. In particular, it is shown that there exist countable perfect sets in the Ellentuck topology.

How to cite

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Reardon, Patrick. "Ramsey, Lebesgue, and Marczewski sets and the Baire property." Fundamenta Mathematicae 149.3 (1996): 191-203. <http://eudml.org/doc/212118>.

@article{Reardon1996,
abstract = {We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented.  THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets.  THEOREM. In the Ellentuck topology on $[ω]^ω$, $(s)_0$ is a proper subset of the hereditary ideal associated with (s).  We construct an example in the Ellentuck topology of a set which is first category and measure 0 but which is not $B_r$-measurable. In addition, several theorems concerning perfect sets in the Ellentuck topology are presented. In particular, it is shown that there exist countable perfect sets in the Ellentuck topology.},
author = {Reardon, Patrick},
journal = {Fundamenta Mathematicae},
keywords = {Ramsey set; Marczewski set; perfect set; measurable set; Baire property; density topology; Ellentuck topology; σ-algebra; perfect sets},
language = {eng},
number = {3},
pages = {191-203},
title = {Ramsey, Lebesgue, and Marczewski sets and the Baire property},
url = {http://eudml.org/doc/212118},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Reardon, Patrick
TI - Ramsey, Lebesgue, and Marczewski sets and the Baire property
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 3
SP - 191
EP - 203
AB - We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented.  THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets.  THEOREM. In the Ellentuck topology on $[ω]^ω$, $(s)_0$ is a proper subset of the hereditary ideal associated with (s).  We construct an example in the Ellentuck topology of a set which is first category and measure 0 but which is not $B_r$-measurable. In addition, several theorems concerning perfect sets in the Ellentuck topology are presented. In particular, it is shown that there exist countable perfect sets in the Ellentuck topology.
LA - eng
KW - Ramsey set; Marczewski set; perfect set; measurable set; Baire property; density topology; Ellentuck topology; σ-algebra; perfect sets
UR - http://eudml.org/doc/212118
ER -

References

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  17. [W] J. T. Walsh, Marczewski sets, measure, and the Baire property. II, Proc. Amer. Math. Soc. 106 (4) (1989), 1027-1030. Zbl0671.28002

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