Additivity of measure implies dominating reals
Author:
Arnold W. Miller
Journal:
Proc. Amer. Math. Soc. 91 (1984), 111-117
MSC:
Primary 03E35; Secondary 03E40
DOI:
https://doi.org/10.1090/S0002-9939-1984-0735576-9
MathSciNet review:
735576
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Abstract: We show that additivity of measure , the union of less than continuum many measure zero sets has measure zero) implies that every family
of cardinality less than continuum is eventually dominated (this is the property
). This yields as a corollary from known results that
.
is the property that the union of less than continuum many first category sets has first category and
is the property that the real line is not the union of less than continuum many first category sets. Also, a new property of measure and category is introduced, the covering property,
, which says that for any family of measure zero (first category) sets of cardinality less than the continuum there is some measure zero (first category) set not covered by any member of the family. By dualizing the proof that
we show that
. The weak dominating property,
, says that no small family contained in
dominates every element of
.
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- [3] -, A characterization of the least cardinal for which the Baire category theorem fails, Proc. Amer. Math. Soc. 86 (1982), 498-502. MR 671224 (84b:04002)
- [4] F. Rothberger, Eine Äquivalenz zwischen der Kontinuumhypothese und der Existenz der Lusinschen und Sierpińskischen Mengen, Fund. Math. 30 (1938), 215-217.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1984-0735576-9
Article copyright:
© Copyright 1984
American Mathematical Society