A parallel to the null ideal for inaccessible λ: Part I

It is well known to generalize the meagre ideal replacing ℵ 0 by a (regular) cardinal λ > ℵ 0 and requiring the ideal to be λ +-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing ℵ 0 by λ, so requiring it to be (< λ)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead of forcing) we may look at the Boolean Algebra of λ-Borel sets modulo the ideal. Surprisingly we get a positive = existence answer for λ a " mild " large cardinal: the weakly compact one. We apply this to get consistency results on cardinal invariants for such λ's. We shall deal with other cardinals more properties related forcing notions in a continuation.