A regularity-like property of normal ultrafilters Suppose $\kappa$ is a measurable cardinal, with normal measure $U$.  Is there a sequence $\{ X_\alpha : \alpha < \kappa^+ \} \subseteq U$ such that for every $Y \in  [\kappa^+ ]^\kappa$, $\bigcap_{\alpha \in Y} X_\alpha \notin U$?
 A: Great question! The answer is no.
The main realization here is the following:

Lemma: Suppose $\kappa=\kappa^{<\kappa}$ is regular, $\kappa<\lambda$ is a cardinal, and $X=\{X_\alpha;\alpha<\lambda\}$ is a sequence of subsets of $\kappa$. Then $X$ has a subsequence (of length $\kappa$) that converges to one of the elements of $X$ (in the generalized Baire space topology on $\kappa^\kappa$).

This is true because $\kappa^\kappa$ has a base of size $\kappa$ so not every point of $X$ can be isolated.
Now suppose that $X$ is a sequence of more than $\kappa$ many elements of $U$. Apply the lemma and find a subsequence $Y=\{Y_\alpha;\alpha<\kappa\}$ which converges to some $Z\in U$. Also, let $C$ be the club of those $\alpha$ for which the $\alpha$th initial segment of the elements of $Y$ has stabilized by step $\alpha$; in other words, the $\alpha$ such that $Y_\beta\cap\alpha=Z\cap\alpha$ for all $\beta\geq\alpha$. It is then straightforward to see that $Z\cap C'\cap \triangle Y\subseteq \bigcap_{\alpha\in C\setminus C'} Y_\alpha$ (where $C'$ is the club of limit points of $C$) and so $\bigcap_{\alpha\in C\setminus C'}Y_\alpha\in U$. So $Y$ is a size $\kappa$ subfamily of $X$ whose intersection lies in $U$, meaning that $X$ cannot be as you required.

Your desired property of measures is similar to regularity, but isn't exactly it. Rather, it turns out it has to do with the Tukey ordering. Given posets $A$ and $B$, say that $A\leq_T B$ if there is a map $f\colon B\to A$ that maps dense subsets of $B$ to dense subsets of $A$. This gives rise to equivalence classes, called Tukey classes. For given $\nu$ and $\lambda$ there is always a maximum Tukey class among $\lambda$-directed posets of size $\nu$.
It turns out that your regularity-like property of an ultrafilter on $\kappa$ is equivalent to the ultrafilter representing the top Tukey class of $\kappa$-directed posets of size $\kappa^+$ (or $2^\kappa$, if we don't commit to GCH).
Normal ultrafilters never realize the top Tukey class (in fact, even p-point ultrafilters and their products do not), but nonnormal ones can (given larger cardinals).
