Fine measures, strong compactness and supercompactness

Work in ZFC, with The Axiom of Choice for cardinal sizes, and fix a regular uncountable cardinal kappa as in Cofinality $\mathrm{cf}(\alpha )$, and regular and singular cardinals. For a cardinal lambda>=kappa put

$P_{\kappa }(\lambda )=\{x\subseteq \lambda :\mathrm{\mid }x\mathrm{\mid }<\kappa \}\mathrm{.}$

This is a set by Separation from the power set, and contains empty. A proper kappa-complete ultrafilter U on this index set, in the sense of Complete ultrafilters and measurable cardinals, is fine if $\{x:\alpha \in x\}\in U$ for every alpha<lambda. It is normal if whenever S belongs to U and $f:S\to \lambda$ satisfies f(x) in x for every x in S, some fibre of f belongs to U. In particular such a domain cannot contain the empty index x. Fineness alone does not assert normality.

The cardinal kappa is strongly compact if every proper kappa-complete filter on every set extends to a kappa-complete ultrafilter on that same set. It is lambda-supercompact if P_kappa(lambda) carries a normal fine kappa-complete ultrafilter; it is supercompact if it is lambda-supercompact for every cardinal lambda>=kappa. These are existence properties, not assertions that such cardinals or measures exist. Lambda=kappa is allowed. No comparison between strong compactness and supercompactness is assumed in this definition.