## 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:

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