measurable and real-valued measurable cardinals
is real-valued measurable if there exists a nontrivial -additive measure on .
If is measurable, then it is real-valued measurable. This is so because the ultrafilter and its dual ideal induce a two-valued measure on where every member of is mapped to 1 and every member of is mapped to 0. Since is -complete, is also -complete. It can then be proved that if –the ideal of those sets whose measures are 0–is -complete, then is -additive.
On the converse side, if is not real-valued measurable, then . It can be shown that if is real-valued measurable, then it is regular; a further result is that is weakly inaccessible. Inaccessible cardinals are in some sense ”large.”
|Title||measurable and real-valued measurable cardinals|
|Date of creation||2013-03-22 18:54:53|
|Last modified on||2013-03-22 18:54:53|
|Last modified by||yesitis (13730)|