|Affiliations:||Mathematical Institute of the Serbian Academy of Sciences and Arts||Title:||Von Neumann's problem and large cardinals||Journal:||Bulletin of the London Mathematical Society||Volume:||38||Issue:||6||First page:||907||Last page:||912||Issue Date:||1-Jan-2006||Rank:||M22||ISSN:||0024-6093||DOI:||10.1112/S0024609306018704||Abstract:||
It is a well-known problem of Von Neumann to discover whether the countable chain condition and weak distributivity of a complete Boolean algebra imply that it carries a strictly positive probability measure. It was shown recently by Balcar, Jech and Pazák, and by Veličković, that it is consistent with ZFC, modulo the consistency of a supercompact cardinal, that every ccc weakly distributive complete Boolean algebra carries a contiuous strictly positive submeasure - that is, it is a Maharam algebra. We use some ideas of Gitik and Shelah and implications from the inner model theory to show that some large cardinal assumptions are necessary for this result.
|Publisher:||London Mathematical Society|
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