Normal measure: Difference between revisions
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A '''normal measure''' is a measure on 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. |
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 is a member of the ultrafilter, i.e. has measure 1.) |
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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. |
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== See also == |
== See also == |
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* [[Measurable cardinal]] |
* [[Measurable cardinal]] |
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* [[Club set]] |
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Revision as of 08:57, 7 May 2006
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 is a member of the ultrafilter, i.e. has measure 1.)
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.
See also
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