Generalizing random real forcing for inaccessible cardinals

Abstract

The two parallel concepts of "small" sets of the real line are meagre sets and null sets. Those are equivalent to Cohen forcing and Random real forcing for $\aleph^{\aleph_0}_0$; in spite of this similarity, the Cohen forcing and Random Real forcing have very different shapes. One of these differences is in the fact that the Cohen forcing has an easy natural generalization for $\lambda^{\lambda}$ while $\lambda > \aleph_0$, corresponding to an extension for the meagre sets, while the Random real forcing didn't see to have a natural generalization, as Lebesgue measure doesn't have a generalization for space $\lambda^\lambda$ while $\lambda > \aleph_0$. Shelah found a forcing resembling the properties of Random Real Forcing for $\lambda^\lambda$ while $\lambda$ is a weakly compact cardinal. Here we describe, with additional assumptions, such a forcing for $\lambda^\lambda$ while $\lambda$ is an inaccessible cardinal; this forcing preserves cardinals and cofinalities, however unlike Cohen forcing, does not add undominated reals.