# Normal measure

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In [set theory](/source/Set_theory), a **normal measure** is a measure on a [measurable cardinal](/source/Measurable_cardinal) \kappa such that the equivalence class of the identity function on \kappa maps to \kappa itself in the [ultrapower](/source/Ultraproduct) 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](/source/Diagonal_intersection).

For a normal measure \mu, any [closed unbounded](/source/Club_set) (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

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