Measurable
A measurable cardinal is a certain type of large cardinal which possesses strong properties. It was one of the first large cardinal axioms to be developed, after inaccessible, Mahlo and weakly compact cardinals. Any measurable cardinal is weakly compact, and therefore Mahlo and inaccessible. Furthermore, any measurable cardinal is \(\Pi^2_1\)-indescribable, but not \(\Sigma^2_1\)-indescribable. This doesn't make it weaker than shrewd cardinals, since the correlation between size and consistency strength breaks down at the level of large large cardinals.
Measurable cardinals were the subject of Scott's famous proof that if measurable cardinals existed, then \(V \neq L\): therefore, if a cardinal is measurable, it won't be in \(L\), which can be explained by the fact that the objects necessary to show a measurable cardinal is measurable would not be contained within \(L\).
There are multiple equivalent definitions of measurability, including one in terms of compactness which shows why measurable cardinals are compact.[1] The original definition was that there is a nontrivial ultrafilter on the subsets of \(\kappa\), picking out exactly which subsets of \(\kappa\) are large and which are not, which is closed under \(< \kappa\)-sized intersections. This is a generalization of the existence of a nontrivial ultrafilter on \(\aleph_0\), like how inaccessible cardinals are a generalization of \(\aleph_0\) being inaccessible from finite numbers. If \(\kappa\) is measurable, then "almost all" cardinals below \(\kappa\) are strongly inaccessible, strongly Mahlo, weakly compact, Ramsey and more, in the sense that the set of those is large.