Summary: 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 B. Balcar, T. Jech and T. Pazák [ibid. 37, 885–898 (2005; Zbl 1101.28003)] and by B. Veličković [Isr. J. Math. 147, 209–220 (2005)] 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 M. Gitik and S. Shelah [Isr. J. Math. 124, 221–242 (2001; Zbl 1018.03043)] and implications from the inner model theory to show that some large cardinal assumptions are necessary for this result.