In set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, a normal measure is a measure on a measurable cardinal

Measurable cardinal

- Measurable :Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ...

 κ such that the equivalence class of the identity function on κ maps to κ itself in the ultrapower

Ultraproduct

The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature...

 construction. Equivalently, if f:κ→κ is such that f(α)<α for most α<κ, then there is a β<κ such that f(α)=β for most α<κ. (Here, "most" means that the set of elements of κ where the property holds is a member of the ultrafilter, i.e. has measure 1.) Also equivalent, the ultrafilter (set of sets of measure 1) is closed under diagonal intersection

Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory.If \displaystyle\delta is an ordinal number and \displaystyle\langle X_\alpha \mid \alphaDiagonal intersection is a term used in mathematics, especially in set theory....

.

For a normal measure, any closed unbounded (club) subset of κ contains most ordinals less than κ. And any subset containing most ordinals less than κ is stationary in κ.

If an uncountable cardinal κ has a measure on it, then it has a normal measure on it.