Some Remarks on Normal Measures and Measurable Cardinals

We prove two theorems which in a certain sense show that the number of normal measures a measurable cardinal κ can carry is independent of a given fixed behavior of the continuum function on any set having measure 1 with respect to every normal measure over κ . First, starting with a model V ⊨ “ZFC + GCH + o ( κ ) = δ *” for δ * ≤ κ + any finite or infinite cardinal, we force and construct an inner model N ⊆ V [ G ] so that N ⊨ “ZF + (∀ δ < κ ) [DC δ ] + ¬AC κ + κ carries exactly δ * normal measures + 2 δ = δ ++ on a set having measure 1 with respect to every normal measure over κ ”. There is nothing special about 2 δ = δ here, and other stated values for the continuum function will be possible as well. Then, starting with a model V ⊨ “ZFC + GCH + κ is supercompact”, we force and construct models of AC in which, roughly speaking, regardless of the specified behavior of the continuum function below κ on any set having measure 1 with respect to every normal measure over κ , κ can in essence carry any number of normal measures δ * ≥ κ ++ .