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 } .