measurable and realvalued measurable cardinals
Let $\kappa $ be an uncountable cardinal. Then

1.
$\kappa $ is measurable if there exists a nonprincipal $\kappa $complete^{} ultrafilter $U$ on $\kappa $;

2.
$\kappa $ is realvalued measurable if there exists a nontrivial $\kappa $additive measure $\mu $ on $\kappa $.
If $\kappa $ is measurable, then it is realvalued measurable. This is so because the ultrafilter $U$ and its dual ideal $I$ induce a twovalued measure $\mu $ on $\kappa $ where every member of $U$ is mapped to 1 and every member of $I$ is mapped to 0. Since $U$ is $\kappa $complete, $I$ is also $\kappa $complete. It can then be proved that if ${I}_{\mu}$–the ideal of those sets whose measures are 0–is $\kappa $complete, then ${I}_{\mu}$ is $\kappa $additive.
On the converse^{} side, if $\kappa $ is not realvalued measurable, then $\kappa \le {2}^{{\mathrm{\aleph}}_{0}}$. It can be shown that if $\kappa $ is realvalued measurable, then it is regular^{}; a further result is that $\kappa $ is weakly inaccessible. Inaccessible cardinals are in some sense ”large.”
Title  measurable and realvalued measurable cardinals 

Canonical name  MeasurableAndRealvaluedMeasurableCardinals 
Date of creation  20130322 18:54:53 
Last modified on  20130322 18:54:53 
Owner  yesitis (13730) 
Last modified by  yesitis (13730) 
Numerical id  4 
Author  yesitis (13730) 
Entry type  Definition 
Classification  msc 03E55 