Applications of pcf for mild large cardinals to elementary embeddings

The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing sequence (lambda_i | i < kappa) of regular cardinals converging to mu such that lambda = tcf(prod_{i < kappa} lambda_i, <_{J^{bd}_kappa}). 2. Let mu be a strong limit cardinal and theta a cardinal above mu. Suppose that at least one of them has an uncountable cofinality. Then there is sigma_* < mu such that for every chi < theta the following holds: theta > sup{sup pcf_{sigma_*-complete} (frak a) | frak a subseteq Reg cap (mu^+, chi) and |frak a| < mu}. As an application we show that: if kappa is a measurable cardinal and j:V to M is the elementary embedding by a kappa-complete ultrafilter over kappa, then for every tau the following holds: 1. if j(tau) is a cardinal then j(tau) = tau; 2. |j(tau)| = |j(j(tau))|; 3. for any kappa-complete ultrafilter W on kappa, |j (tau)| = |j_W(tau)|. The first two items provide affirmative answers to questions from Gitik and Shelah (1993) [2] and the thrid to a question of D. Fremlin.