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On a problem of Banach

Authors: Andrzej Pelc and Karel Prikry
Journal: Proc. Amer. Math. Soc. 89 (1983), 608-610
MSC: Primary 28C10; Secondary 04A20, 04A30
MathSciNet review: 718982
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Abstract: Assuming the continuum hypothesis, we obtain a translation invariant version of the following result of E. Grzegorek: There are two countably generated $ \sigma $-algebras on the interval [0, 1] such that both carry a nonatomic countably additive probability measure, but the $ \sigma $-algebra generated by their union does not carry any such measure.

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  • [1] S. Banach, Sur les suites d'ensembles excluant l'existence d'une mesure (Note posthume avec préface et commentaire de E. Marczewski), Colloq. Math. 1 (1948), 103-108. MR 0025547 (10:23e)
  • [2] E. Grzegorek, Remarks on $ \sigma $-fields without continuous measures, Colloq. Math. 39 (1978), 73-75. MR 507265 (80b:28002)
  • [3] -, Solution of a problem of Banach on $ \sigma $-fields without continuous measure, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 7-10. MR 616191 (82h:04005)
  • [4] S. Kakutani and J. Oxtoby, Construction of a non-separable invariant extension of the Lebesgue measure space, Ann. of Math. (2) 52 (1950), 580-590. MR 0037335 (12:246g)
  • [5] B. V. Rao, On discrete Borel spaces and protective sets, Bull. Amer. Math. Soc. 75 (1969), 614-617. MR 0242684 (39:4014)
  • [6] W. Sierpiński, Sur les translations des ensembles linéaires, Fund. Math. 19 (1932), 22-28.
  • [7] -, Hypothese du continu, Chelsea, New York, 1956.

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Keywords: Translation invariant measure, countably generated $ \sigma $-algebra, continuum hypothesis
Article copyright: © Copyright 1983 American Mathematical Society

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