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We prove that o( )= is sucient to construct a model V (C )i n which is measurable and C is a closed and unbounded subset of containing only inaccessible cardinals of V. Gitik proved that o( )= is necessary. We also calculate the consistency strength of the existence of such a set C together with the assumption that is Mahlo, weakly compact, or Ramsey. In addition we consider the possibility of having the set C generate the closed unbounded ultralter of V while remains measurable, and show that Radin forcing, which requires a weak repeat point, cannot be improved on.

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