A null ideal for inaccessibles

Abstract

In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of $2^\kappa$, $\kappa$ inaccessible, and study its associated ideal of null sets and notion of measurability. This issue was addressed by Shelah (On CON(Dominating$\_$lambda$\,>\,$cov$\_\lambda$(meagre)), arXiv:0904.0817, Problem 0.5) and concerns the definition of a forcing which is $\kappa ^\kappa$-bounding, $<\kappa$-closed and $\kappa ^+$-cc, for $\kappa$ inaccessible. Cohen and Shelah (Generalizing random real forcing for inaccessible cardinals, arXiv:1603.08362) provide a proof for (Shelah, On CON(Dominating$\_$lambda$\,>\,$cov$\_\lambda$(meagre)), arXiv:0904.0817, Problem 0.5), and in this paper we independently reprove this result by using a different type of construction. This also contributes to a line of research adressed in the survey paper (Khomskii et al. in Math L Q 62(4–5):439–456, 2016).