ℙ max variations related to slaloms

We prove the iteration lemmata, which are the key lemmata to show that extensions by Pmax variations satisfy absoluteness for Π 2 ‐statements in the structure 〈 H ( ω 2 ), ∈, NS ω 1 , R 〉 for some set R of reals in L (ℝ), for the following statements: (1) The cofinality of the null ideal is ℵ 1 . (2) There exists a good basis of the strong measure zero ideal. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)