Fine ultrafilter ($\{x:\alpha \in x\}\in U$)

An ultrafilter on $P_{\kappa }(\lambda )$ is fine when it contains the set of small subsets containing $\alpha$, for each $\alpha <\lambda$. Fineness makes every coordinate of $\lambda$ represented in the corresponding ultrapower, though no one small subset is required to contain every coordinate.

An ultrafilter $U$ on $P_{\kappa }(\lambda )$ is normal when each function with $f(x)\in x$ on a $U$-large set is constant on a $U$-large subset. This is the small-subset version of normality; together with fineness and kappa-completeness it gives the measure characterization of a supercompact cardinal.