We reprove Gitik’s theorem that if the GCH holds and
$o(\gk)=\gk+1$ then there is a generic extension in which $\gk$ is
still measurable and there is a closed unbounded subset C of $\gk$
such that every $ν\in C$ is inaccessible in the ground model.
Unlike the forcing used by Gitik, the iterated forcing
$\radin\gl+1$ used in this paper has the property that if $\gl$
is a cardinal less then $\gk$ then $\radin\gl+1$ can be factored
in V as $\radin\gk+1=\radin\gl+1\times\radin\gl+1,\gk$
where $\card{\radin\gl+1}\le\gl+$ and $\radin\gl+1,\gk$ does
not add any new subsets of $\gl$.