Some non-normal κ-complete ultrafilters over a measurable κ with special properties are constructed. Questions by A. Kanamori [4] about infinite Rudin-Frolik sequences, discreteness and products are answered.

We deal with some natural properties of ultrafilters which trivially fail for normal ultrafilters.

Gitik, M., On measurable cardinals violating the continuum hypothesis, Annals of Pure and Applied Logic 63 227-240. It is shown that an extender used uncountably many times in an iteration is reconstructible. This together with the Weak Covering Lemma is used to show that the assumption o=κ+α is necessary for a measurable κ with 2κ=κ+α.

A method of iteration of Prikry type forcing notions as well as a forcing for adding clubs is presented. It is applied to construct a model with a measurable cardinal containing a club of former regulars, starting with o(κ) = κ + 1. On the other hand, it is shown that the strength of above is at least o(κ) = κ.