More on real-valued measurable cardinals and forcing with ideals

(1) It is shown that if c is real-valued measurable then the Maharam type of (c, P(c), σ) is 2^c. This answers a question of D. Fremlin [Fr, (P2f)]. (2) A different construction of a model with a real-valued measurable cardinal is given from that of R. Solovay [So]. This answers a question of D. Fremlin [Fr, (P1)]. (3) The forcing with a K-complete ideal over a set X, |X| ≥ K cannot be isomorphic to Random x Cohen or Cohen x Random. The result for X = K was proved in [Gi-Sh1] but, as was pointed out to us by M. Burke, the application of it in [Gi-Sh2] requires dealing with any X. The application is: if A_n is a set of reals for n < ω then for some pairwise disjoint B_n (for n < ω) we have B_n ⊆ A_n but they have the same outer Lebesgue measure.