Filters and sequences

Sławomir Solecki

Fundamenta Mathematicae (2000)

  • Volume: 163, Issue: 3, page 215-228
  • ISSN: 0016-2736

Abstract

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We consider two situations which relate properties of filters with properties of the limit operators with respect to these filters. In the first one, we show that the space of sequences having limits with respect to a Π 3 0 filter is itself Π 3 0 and therefore, by a result of Dobrowolski and Marciszewski, such spaces are topologically indistinguishable. This answers a question of Dobrowolski and Marciszewski. In the second one, we characterize universally measurable filters which fulfill Fatou’s lemma.

How to cite

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Solecki, Sławomir. "Filters and sequences." Fundamenta Mathematicae 163.3 (2000): 215-228. <http://eudml.org/doc/212440>.

@article{Solecki2000,
abstract = {We consider two situations which relate properties of filters with properties of the limit operators with respect to these filters. In the first one, we show that the space of sequences having limits with respect to a $Π^0_3$ filter is itself $Π^0_3$ and therefore, by a result of Dobrowolski and Marciszewski, such spaces are topologically indistinguishable. This answers a question of Dobrowolski and Marciszewski. In the second one, we characterize universally measurable filters which fulfill Fatou’s lemma.},
author = {Solecki, Sławomir},
journal = {Fundamenta Mathematicae},
keywords = {filters; separation property; Fatou's lemma},
language = {eng},
number = {3},
pages = {215-228},
title = {Filters and sequences},
url = {http://eudml.org/doc/212440},
volume = {163},
year = {2000},
}

TY - JOUR
AU - Solecki, Sławomir
TI - Filters and sequences
JO - Fundamenta Mathematicae
PY - 2000
VL - 163
IS - 3
SP - 215
EP - 228
AB - We consider two situations which relate properties of filters with properties of the limit operators with respect to these filters. In the first one, we show that the space of sequences having limits with respect to a $Π^0_3$ filter is itself $Π^0_3$ and therefore, by a result of Dobrowolski and Marciszewski, such spaces are topologically indistinguishable. This answers a question of Dobrowolski and Marciszewski. In the second one, we characterize universally measurable filters which fulfill Fatou’s lemma.
LA - eng
KW - filters; separation property; Fatou's lemma
UR - http://eudml.org/doc/212440
ER -

References

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  1. [DM] T. Dobrowolski and W. Marciszewski, Classification of function spaces with the pointwise topology determined by a countable dense set, Fund. Math. 148 (1995), 35-62. Zbl0834.46016
  2. [K] A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995. 
  3. [L] A. Louveau, Sur la génération des fonctions boréliennes fortement affines sur un convexe compact métrisable, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 2, 57-68. Zbl0604.46012
  4. [M] K. Mazur, F σ -ideals and ω 1 ω 1 * -gaps in the Boolean algebra P(ω)/I, Fund. Math. 138 (1991), 103-111. 
  5. [R] H. L. Royden, Real Analysis, Macmillan, 1988. Zbl0704.26006

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