A real-valued measurable cardinal is an uncountable cardinal 
carrying a probability measure on all of its
subsets that vanishes on singletons
and is additive over disjoint families of size less than 
. The measure can take arbitrary real values in ![[0,1]](/images/equations/Real-ValuedMeasurableCardinal/Inline3.svg)
, rather than only 0 and 1 as in the ultrafilter
formulation of a measurable cardinal.
Real-Valued Measurable Cardinal
See also
Cardinal Number, Measurable Cardinal, Probability MeasureExplore 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. "Real-Valued Measurable Cardinal." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Real-ValuedMeasurableCardinal.html