Generic embeddings associated to an indestructibly weakly compact cardinal

I use generic embeddings induced by generic normal measures on P κ ( λ ) that can be forced to exist if κ is an indestructibly weakly compact cardinal. These embeddings can be applied in order to obtain the forcing axioms MA + + ( < μ -closed ) in forcing extensions. This has consequences in V : The Singular Cardinal Hypothesis holds above κ , and κ has a useful Jónsson-like property. This in turn implies that the countable tower Q < κ works much like it does when κ is a Woodin limit of Woodin cardinals. One consequence is that every set of reals in the Chang model is Lebesgue measurable and has the Baire Property, the Perfect Set Property and the Ramsey Property. So indestructible weak compactness has effects on cardinal arithmetic high up and also on the structure of sets of real numbers, down low, similar to supercompactness.