= Solution
Fix a regular uncountable <cardinal number> $\kappa$. A <club set> $C\subseteq\kappa$ is unbounded in $\kappa$ and closed under suprema below $\kappa$ of nonempty increasing sequences with no last term. Equivalently, every <limit ordinal> $\delta<\kappa$ with $\sup(C\cap\delta)=\delta$ belongs to $C$. The <club filter> is
$$
\boxed{\mathcal D_\kappa=\{A\subseteq\kappa:\exists C\subseteq A\text{ with }C\text{ club in }\kappa\}}.
$$
It consists of all sets containing a <club set>, not only the club sets themselves.
An <omega-measurable cardinal> is a <cardinal number> carrying a <nonprincipal ultrafilter> closed under countable intersections, that is, a <countably complete ultrafilter>. Such a cardinal must be uncountable: on a countable set the intersection of the complements of all singletons would be empty.
A <measurable cardinal> is an uncountable <cardinal number> $\kappa$ carrying a nonprincipal $\kappa$-complete <ultrafilter>. A <kappa-complete filter> is closed under intersections of every family of size strictly below $\kappa$. Thus for an <omega-measurable cardinal> the completeness required is countable completeness, not merely closure under finitely many intersections.
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