Normal measure
In set theory, a normal measure is a measure on a measurable cardinal κ such that the equivalence class of the identity function on κ maps to κ itself in the ultrapower construction. Equivalently, if f:κ→κ is such that f<α for most α<κ, then there is a β<κ such that f=β for most α<κ. Also equivalent, the ultrafilter is closed under diagonal intersection.
For a normal measure, any closed unbounded subset of κ contains most ordinals less than κ. And any subset containing most ordinals less than κ is stationary in κ.
If an uncountable cardinal κ has a measure on it, then it has a normal measure on it.
