Indestructibility and destructible measurable cardinals

Say that ({kappa})’s measurability is destructible if there exists a < ({kappa})-closed forcing adding a new subset of ({kappa}) which destroys ({kappa})’s measurability. For any δ, let λ_δ =_df The least beth fixed point above δ. Suppose that ({kappa}) is indestructibly supercompact and there is a measurable cardinal λ > ({kappa}). It then follows that ({A_{1} = {delta < kappa mid delta}) is measurable, δ is not a limit of measurable cardinals, δ is not δ⁺ strongly compact, and δ’s measurability is destructible when forcing with partial orderings having rank below λ_δ} is unbounded in ({kappa}). On the other hand, under the same hypotheses, ({A_{2} = {delta < kappa mid delta}) is measurable, δ is not a limit of measurable cardinals, δ is not δ⁺ strongly compact, and δ′s measurability is indestructible when forcing with either Add(δ, 1) or Add(δ, δ⁺)} is unbounded in ({kappa}) as well. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two distinct models in which either ({A_{1} = emptyset}) or ({A_{2} = emptyset}). In each of these models, both of which have restricted large cardinal structures above ({kappa}), every measurable cardinal δ which is not a limit of measurable cardinals is δ⁺ strongly compact, and there is an indestructibly supercompact cardinal ({kappa}). In the model in which ({A_{1} = emptyset}), every measurable cardinal δ which is not a limit of measurable cardinals is <λ_δ strongly compact and has its <λ_δ strong compactness (and hence also its measurability) indestructible when forcing with δ-directed closed partial orderings having rank below λ_δ. The choice of the least beth fixed point above δ is arbitrary, and other values of λ_δ are also possible.