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On measures on complete Boolean algebras

  • Karel Prikry (a1)

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In this paper we prove some theorems concerning measures on complete Boolean algebras. Among other things, in §I of this paper, we construct a counterexample to the following conjecture of W. Luxemburg: Every measure on a nonatomic hyperstonian Boolean algebra is normal. (See [3, p. 57].) This result is expressed by Theorem 1, §I. In order to construct this example we have to suppose that a real-valued measurable cardinal exists. This hypothesis is independent of the usual axioms of set theory. Luxemburg proved that our assumption is necessary. Our second result is stated in Theorem 2 near the end of the paper.

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[1]Halmos, P. R., Measure theory, D. Van Nostrand Co., New York, 1950.
[2]Keisler, H. J. and Tarski, A., From accessible to inaccessible cardinals, Fundamenta Mathematicae, vol. 53 (1964), pp. 225308.
[3]Luxemburg, W. A. J., On the existence of σ-complete prime ideals in Boolean algebras, Colloquium Mathematicum, vol. 19 (1968), pp. 5158.
[4]Prikry, K., On σ-complete prime ideals, Colloquium Mathematicum, vol. XXII (1971), pp. 209214.
[5]Prikry, K., Changing measurable into accessible cardinals, Dissertationes Mathematicae, vol. 68 (1970), pp. 552.
[6]Sikorski, R., Boolean algebras, Springer-Verlag, Berlin, 1964.

On measures on complete Boolean algebras

  • Karel Prikry (a1)

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