set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

, a normal measure is a measure on a

measurable cardinal In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivisi ...

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

ultrapower The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factor ...

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

. 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.

References

* {{cite book, last=Kanamori, first=Akihiro, authorlink=Akihiro Kanamori, year=2003, publisher=Springer , title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings, title-link= The Higher Infinite , edition=1st , isbn=3-540-57071-3 pp 52–53 Large cardinals Measures (set theory)

See also

References