A null ideal for inaccessibles

In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of 2 ^κ, κ inaccessible, and study its associated ideal of null sets and notion of measurability. This issue was addressed by Shelah (On CON(Dominating_lambda>cov_λ(meagre)), arXiv:0904.0817, Problem 0.5) and concerns the definition of a forcing which is κ^κ-bounding, < κ-closed and κ⁺-cc, for κ inaccessible. Cohen and Shelah (Generalizing random real forcing for inaccessible cardinals, arXiv:1603.08362) provide a proof for (Shelah, On CON(Dominating_lambda>cov_λ(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).