Authors: Farah, Ilijas 
Veličković, Boban
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
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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