On an extendability problem for measures (Q760703)

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.