Normal measure

In set theory, a normal measure is a measure on a measurable cardinal ${\displaystyle \kappa }$ such that the equivalence class of the identity function on ${\displaystyle \kappa }$ maps to ${\displaystyle \kappa }$ itself in the ultrapower construction. Equivalently, a measure ${\displaystyle \mu }$ on ${\displaystyle \kappa }$ is normal iff whenever ${\displaystyle f:\kappa \to \kappa }$ is such that ${\displaystyle f(\alpha )<\alpha }$ for ${\displaystyle \mu }$-many ${\displaystyle \alpha <\kappa }$, then there is a ${\displaystyle \beta <\kappa }$ such that ${\displaystyle f(\alpha )=\beta }$ for ${\displaystyle \mu }$-many ${\displaystyle \alpha <\kappa }$. (Here, "${\displaystyle \mu }$-many" means that the set of elements of ${\displaystyle \kappa }$ where the property holds is a member of the ultrafilter, i.e. has measure 1 in ${\displaystyle \mu }$.) Also equivalent, the ultrafilter (set of sets with measure 1) is closed under diagonal intersection.

For a normal measure ${\displaystyle \mu }$, any closed unbounded (club) subset of ${\displaystyle \kappa }$ contains ${\displaystyle \mu }$-many ordinals less than ${\displaystyle \kappa }$ and any subset containing ${\displaystyle \mu }$-many ordinals less than ${\displaystyle \kappa }$ is stationary in ${\displaystyle \kappa }$.

If an uncountable cardinal ${\displaystyle \kappa }$ has a measure on it, then it has a normal measure on it.