Abstract(1) It is shown that ifc is real-valued measurable then the Maharam type of (c, P(c),σ) is 2c. 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 aκ-complete ideal over a setX, |X| ≥κ cannot be isomorphic to Random × Cohen or Cohen × Random. The result forX=κ 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 anyX. The application is: ifAn is a set of reals forn<ω then for some pairwise disjointBn (forn<ω) we haveBn ⊆An but they have the same outer Lebesgue measure.
[1]
Robert M. Solovay,et al.
Real-valued measurable cardinals
,
1967
.
[2]
Saharon Shelah,et al.
Forcings with ideals and simple forcing notions
,
1989
.
[3]
Saharon Shelah.
Further cardinal arithmetic
,
1996
.
[4]
Karel Prikry.
Ideals and powers of cardinals
,
1975
.
[5]
Saharon Shelah.
How special are Cohen and random forcings, i.e. boolean algebras of the family of subsets of reals modulo meagre or null
,
1993
.
[6]
Saharon Shelah,et al.
More on Simple Forcing Notions and Forcings with Ideals
,
1993,
Ann. Pure Appl. Log..