Easton's theorem and large cardinals from the optimal hypothesis

The equiconsistency of a measurable cardinal with Mitchell order o(κ) = κ++ with a measurable cardinal such that 2κ = κ++ follows from the results by W. Mitchell [13] and M. Gitik [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ (see [8]). In [5], we formulated and proved Easton’s theorem [4] in a large cardinal setting, using slightly stronger hypotheses than the lower bounds identified by Mitchell and Gitik (we used the assumption that the relevant target model contains H(μ), for a suitable μ, instead of the cardinals with the appropriate Mitchell order). In this paper, we use a new idea which allows us to carry out the constructions in [5] from the optimal hypotheses. It follows that the lower bounds identified by Mitchell and Gitik are optimal also with regard to the general behaviour of the continuum function on regulars in the context of measurable cardinals.