Normal measure: Difference between revisions
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== References == |
== References == |
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* {{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|edition=1st |isbn=3-540-57071-3}} pp 52–53 |
* {{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|url=https://archive.org/details/higherinfinitela0000kana|url-access=registration|edition=1st |isbn=3-540-57071-3}} pp 52–53 |
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[[Category:Large cardinals]] |
[[Category:Large cardinals]] |
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Revision as of 19:22, 20 November 2019
In set theory, a normal measure is a measure on a measurable cardinal κ such that the equivalence class of the identity function on κ maps to κ itself in the ultrapower 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.
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.
See also
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