A measurable cardinal with a closed unbounded set of inaccessibles from $o(\kappa)=\kappa$

We prove that $o(\kappa)=\kappa$ is sufficient to construct a model $V[C]$in which $\kappa$ is measurable and $C$ is a closed and unbounded subset of $\kappa$ containing only inaccessible cardinals of $V$. Gitik proved that $o(\kappa)=\kappa$ is necessary.

We also calculate the consistency strength of the existence of such a set $C$ together with the assumption that $\kappa$ is Mahlo, weakly compact, or Ramsey. In addition we consider the possibility of having the set $C$ generate the closed unbounded ultrafilter of $V$ while $\kappa$ remains measurable, and show that Radin forcing, which requires a weak repeat point, cannot be improved on.