Preserving preservation

We prove that the property " P doesn't make the old reals Lebesgue null " is preserved under countable support iterations of proper forcings, under the additional assumption that the forcings are nep (a generalization of Suslin proper) in an absolute way. We also give some results for general Suslin ccc ideals. Hypothesis 1: Let (P β , ˜ Q β) β<< be a countable support iteration of proper forcings (a limit) such that each P β (β <) forces that the set of old reals X V ∩ 2 ω remains Lebesgue positive. Then P forces this as well. The main result of this paper (9.4) is that hypothesis 1 is true under some additional (relatively mild) requirements on the P β. It seems that such requirements are needed (this is argued in section 4). Preservation theorems of this kind have proven to be extremely useful in independence proofs. Hypothesis 1 specifically is used in the proof of the following two theorems of [9]: It cannot be decided in ZFC whether every superposition-measurable function is measurable. Forcing is a very general method for proving independence results, i.e., results of the form " formula ϕ is neither provable nor refutable in ZFC ". Forcing gives a method for modifying a given set-theoretical universe V to a new universe V in which some formula ψ is guaranteed to hold. In a forcing argument (for example, to violate CH) one typically has to The authors thank a referee for proposing numerous enhancements, including a substantial simplification of lemma 5.11.