Easton’s theorem and large cardinals

The continuum function α↦2α on regular cardinals is known to have great freedom. Let us say that F is an Easton function iff for regular cardinals α and β, and α<β→F(α)≤F(β). The classic example of an Easton function is the continuum function α↦2α on regular cardinals. If GCH holds then any Easton function is the continuum function on regular cardinals of some cofinality-preserving extension V[G]; we say that F is realised in V[G]. However if we also wish to preserve measurable cardinals, new restrictions must be put on F. We say that κ is F(κ)-hypermeasurable iff there is an elementary embedding j:V→M with critical point κ such that H(F(κ))V⊆M; j will be called a witnessing embedding. We will show that if GCH holds then for any Easton function F there is a cofinality-preserving generic extension V[G] such 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 V[G].