measurable and real-valued 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 real-valued measurable if there exists a nontrivial $\kappa$-additive measure $\mu$ on $\kappa$.

If $\kappa$ is measurable, then it is real-valued measurable. This is so because the ultrafilter $U$ and its dual ideal $I$ induce a two-valued 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 real-valued measurable, then $\kappa \le 2^{{\mathrm{\aleph }}_0}$. It can be shown that if $\kappa$ is real-valued measurable, then it is regular; a further result is that $\kappa$ is weakly inaccessible. Inaccessible cardinals are in some sense ”large.”