  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.
