Normal measure
In set theory, a normal measure is a measure on a measurable cardinal such that the equivalence class of the identity function on maps to itself in the ultrapower construction. Equivalently, a measure on is normal iff whenever is such that for -many, then there is a such that for -many. Also equivalent, the ultrafilter is closed under diagonal intersection.
For a normal measure, any closed unbounded subset of contains -many ordinals less than and any subset containing -many ordinals less than is stationary in.
If an uncountable cardinal has a measure on it, then it has a normal measure on it.