In set theory, a normal measure is a measure on a measurable cardinal \kappa such that the equivalence class of the identity function on \kappa maps to \kappa itself in the ultrapower construction. Equivalently, a measure \mu on \kappa is normal iff whenever f:\kappa\to\kappa is such that f(\alpha)<\alpha for \mu-many \alpha<\kappa, then there is a \beta<\kappa such that f(\alpha)=\beta for \mu-many \alpha<\kappa. (Here, "\mu-many" means that the set of elements of \kappa where the property holds is a member of the ultrafilter, i.e. has measure 1 in \mu.) Also equivalent, the ultrafilter (set of sets with measure 1) is closed under diagonal intersection.
For a normal measure \mu, any closed unbounded (club) subset of \kappa contains \mu-many ordinals less than \kappa and any subset containing \mu-many ordinals less than \kappa is stationary in \kappa.
If an uncountable cardinal \kappa has a measure on it, then it has a normal measure on it.
References
- Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings. 1st ed. Springer. ISBN 3-540-57071-3. pp 52–53