A measurable cardinal is an uncountable cardinal 
admitting a nonprincipal 
-complete ultrafilter on
.
Here 
-complete
means closure under intersections of fewer than
members, and nonprincipal means that no singleton
belongs to the ultrafilter. Equivalently, it supports
a 
-valued
probability measure on all subsets,
vanishing on singletons and additive over disjoint
families of size less than 
.
Measurable Cardinal
See also
Cardinal Number, Real-Valued Measurable Cardinal, UltrafilterExplore with Wolfram|Alpha
References
Jech, T. Set Theory: The Third Millennium Edition, Revised and Expanded. Berlin, Germany: Springer, 2003. https://doi.org/10.1007/3-540-44761-X.Cite this as:
Weisstein, Eric W. "Measurable Cardinal." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MeasurableCardinal.html