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).