More on simple forcing notions and forcings with ideals

(1) It is shown that cardinals below a real-valued measurable cardinal can be split into finitely many intervals so that the powers of cardinals from the same interval are the same. This generalizes a theorem of Prikry [9]. (2) Suppose that the forcing with a κ-complete ideal over κ is isomorphic to the forcing of λ-Cohen or random reals. Then for some τ<κ, λ^τ≥2^κ and λ≤2^<κ implies that 2^κ=2^τ= cov(λ, κ, τ⁺, 2). In particular, if 2^κ<κ^+ω, then λ=2^κ. This answers a question from [3]. (3) If A₀, A₁,..., A_n,... are sets of reals, then there are disjoint sets B₀, B₁,..., B_n,... such that B_n⊆A_n and μ*(B_n)=μ*(A_n) for every n<ω, where μ* is the Lebes gue outer measure. For finitely many sets the result is due to N. Lusin. (4) Let (P, ≤) be a σ-centered forcing notion and 〈A_n |n<ω〉 subsets of P witnessing this. If P, A_n's and the relation of compatibility are Borel, then P adds a Cohen real. (5) The forcing with a κ-complete ideal over a set X, |X|≥κ cannot be isomorphic to a Hechler real forcing. This result was claimed in [3], but the proof given there works only for X of cardinality κ.