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On an extendability problem for measures. (English) Zbl 0555.60003

The author gives a sufficient condition for the existence of extensions of probability measures from a certain class of \(\sigma\)-algebras to countable extensions of these \(\sigma\)-algebras. Then he uses this result to show: Assume that there is no measurable cardinal \(\leq 2^{\aleph_ 0}\). Then there exists a probability space (\(\Omega\),\({\mathcal A},\mu)\) such that whenever \({\mathcal B}\) is a countably generated \(\sigma\)-algebra on \(\Omega\), \(\mu\) can be extended to a measure on \(\sigma\) (\({\mathcal A}\cup {\mathcal B})\), but \(\mu\) has no extension to a measure on \({\mathcal P}(\Omega)\). This answers a question posed some years ago by C. Ryll- Nardzewski under an additional but fairly weak set-theoretic assumption in the negative.

MSC:

60A10 Probabilistic measure theory
28A12 Contents, measures, outer measures, capacities
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References:

[1] ASCHERL, A., LEHN, J.: Two principles for extending probability measures. Manuscripta math.21, 43-50 (1977) · Zbl 0368.60005
[2] JECH, T.: ?Set Theory?, Pure and Applied Mathematics, Academic Press, New York, 1978 · Zbl 0419.03028
[3] JERSCHOW (ERSHOV), M.P.: Extension of measures and stochastic equations. Theor. Probability Appl.19, 431-444 (1974) · Zbl 0312.28001
[4] SCOTT, D.: Measurable cardinals and constructible sets, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys.9, 521-524 (1961) · Zbl 0154.00702
[5] SOLOVAY, R.: Real-valued measurable cardinals, in: ?Axiomatic Set Theory?, Proc. Symp. Pure Math.13, I (D. Scott, ed), 397-428, Amer. Math. Soc., Providence, 1971
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