A regularity-like property of normal ultrafilters

Question

Suppose $\kappa$ is a measurable cardinal, with normal measure $U$. Is there a sequence $\{ X_\alpha : \alpha < \kappa^+ \} \subseteq U$ such that for every $Y \in [\kappa^+ ]^\kappa$, $\bigcap_{\alpha \in Y} X_\alpha \notin U$?

Answer 1

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.

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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).