### Generalizing random real forcing for inaccessible cardinals

by Cohen and Shelah. [CnSh:1085]

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 . In work [1], 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 dominating
reals.

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