Easton's theorem and large cardinals

The continuum function F on regular cardinals is known to have great freedom; if α, β are regular cardinals, then F needs only obey the following two restrictions: (1) cf(F (α)) > α, (2) α < β → F (α) ≤ F (β). However, if we wish to preserve measurable cardinals in the generic extension, new restrictions must be put on F . We say that κ is F (κ)-hypermeasurable if there is an elementary embedding j : V →M with critical point κ such that H(F (κ))V ⊆ M ; j will be called the witnessing embedding. We will show that if κ, closed under F , is F (κ)-hypermeasurable in V and there is a witnessing embedding j such that j(F )(κ) ≥ F (κ), then κ will remain measurable in some generic extension realizing F .