Let $\kappa$ be a measurable cardinal with ultrafilter $U$.  Call sets in $U$ "large"; others, small.   Let $X$ be the set of all finite sequences from $\kappa$.  We call a subsets $A \subseteq X$ closed if it has the following property: 

* Whenever $s$ in $X$, and almost all successors of $s$ are in $A$, then also $s$ is in $A$: 

More precisely: If the set $\{i \in \kappa: (s,i)\in A\}$is in $U$, then also $s\in A$.