Normal measure

In set theory, a normal measure is a measure on a measurable cardinal κ {\displaystyle \kappa } such that the equivalence class of the identity function on κ {\displaystyle \kappa } maps to κ {\displaystyle \kappa } itself in the ultrapower construction. Equivalently, a measure μ {\displaystyle \mu } on κ {\displaystyle \kappa } is normal iff whenever f : κ → κ {\displaystyle f:\kappa \to \kappa } is such that f ( α ) < α {\displaystyle f(\alpha )<\alpha } for μ {\displaystyle \mu } -many α < κ {\displaystyle \alpha <\kappa } , then there is a β < κ {\displaystyle \beta <\kappa } such that f ( α ) = β {\displaystyle f(\alpha )=\beta } for μ {\displaystyle \mu } -many α < κ {\displaystyle \alpha <\kappa } . (Here, " μ {\displaystyle \mu } -many" means that the set of elements of κ {\displaystyle \kappa } where the property holds is a member of the ultrafilter, i.e. has measure 1 in μ {\displaystyle \mu } .) Also equivalent, the ultrafilter (set of sets with measure 1) is closed under diagonal intersection. For a normal measure μ {\displaystyle \mu } , any closed unbounded (club) subset of κ {\displaystyle \kappa } contains μ {\displaystyle \mu } -many ordinals less than κ {\displaystyle \kappa } and any subset containing μ {\displaystyle \mu } -many ordinals less than κ {\displaystyle \kappa } is stationary in κ {\displaystyle \kappa } . If an uncountable cardinal κ {\displaystyle \kappa } has a measure on it, then it has a normal measure on it.