Closed measure zero sets.

*(English)*Zbl 0764.03018Summary: We study the relationship between the \(\sigma\)-ideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal of closed measure zero sets is equal to the additivity of the ideal of meager sets.

##### MSC:

03E35 | Consistency and independence results |

03E05 | Other combinatorial set theory |

28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |

##### Keywords:

null sets; \(\sigma\)-ideal generated by closed measure zero sets; meager sets; category; additivity
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\textit{T. Bartoszynski} and \textit{S. Shelah}, Ann. Pure Appl. Logic 58, No. 2, 93--110 (1992; Zbl 0764.03018)

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##### References:

[1] | Bartoszynski, T., Additivity of measure implies additivity of category, Trans. AMS, 1, (1984) · Zbl 0538.03042 |

[2] | Bartoszynski, T., Combinatorial aspects of measure and category, Fund. math., (1987) · Zbl 0635.04001 |

[3] | Bartoszynski, T., On covering of the real line by null sets, Pacific J. math., 1, (1988) · Zbl 0643.03034 |

[4] | Bartoszynski, T.; Judah, H., On the cofinality of the smallest covering of the real line by meager sets, J. symbolic logic, 54, 3, 828-832, (1989) · Zbl 0686.03023 |

[5] | Bartoszynski, T.; Judah, H., Jumping with random reals, Ann. pure appl. logic, 48, 197-213, (1990) · Zbl 0711.03021 |

[6] | Fremlin, D.; Miller, A., On some properties of Hurewicz Menger and rothberger, Fund. math., 129, (1988) · Zbl 0665.54026 |

[7] | Judah, H.; Shelah, S., The kunen-Miller chart, J. symbolic logic, 55, 3, (1990) · Zbl 0718.03037 |

[8] | Miller, A., Some properties of measure and category, Trans. AMS, 266, (1981) · Zbl 0472.03040 |

[9] | Miller, A., Additivity of measure implies dominating reals, Proc. AMS, 91, (1984) · Zbl 0586.03042 |

[10] | J. Oxtoby, Measure and Category (Springer, Berlin). |

[11] | Raisonnier, J.; Stern, J., The strength of measurability hypotheses, Israel J. math., 50, (1985) · Zbl 0602.03012 |

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