Name of infinite cardinals which has nonprincipal $\sigma$-complete ultrafilters?

Question

The book "General Topology" by Engelking defines non-measurable cardinals as cardinals admitting no nonprincipal $\sigma$-complete ultrafilters. And then it claims that the discrete space of size $\kappa \ge \aleph_0$ is realcompact iff $\kappa$ is non-measurable. It can be proven that the discrete space of size $\kappa \ge \aleph_0$ is realcompact iff $\kappa$ doesn't admit any nonprincipal $\sigma$-complete ultrafilter, so the claim is true. But I think the definition of measurability is different from what is used widely now. What do we call infinite cardinals which has nonprincipal $\sigma$-complete ultrafilters?

Answer 1

There's no standard name for this property I'm aware of (edit: now I am aware that they are called “Ulam-Measurable” according to Noah’s comment below), but a cardinal has this property if and only if measurable cardinals exist and it is greater than or equal to the least measurable cardinal. (Where a cardinal $\kappa$ is called measurable if it has a $\kappa$-complete non-principal ultrafilter.)

1. If $\kappa$ is the least cardinal with a $\sigma$-complete non-principal ultrafilter, and $U$ is such an ultrafilter, then $U$ is $\kappa$-complete: If $\{A_\gamma:\gamma < \lambda\}$ were a partition of $\kappa$ into $\lambda < \kappa$ sets not in $U,$ then we could define $D$ by saying $X\subseteq \lambda$ has $X\in D$ whenever $\bigcup_{\gamma\in X}A_\gamma\in U.$ Then, $D$ would be a non-principal $\sigma$-complete ultrafilter on $\lambda,$ contradicting minimality of $\kappa.$ So $\kappa$ is measurable.
2. If $\lambda>\kappa$ where $\kappa$ is measurable, let $U$ be a $\sigma$-complete nonprincipal ultrafilter on $\kappa.$ Then if we define $D$ by saying $A\subseteq\lambda$ has $A\in D$ whenever $\kappa\cap A\in U,$ then $D$ is a $\sigma$-complete nonprincipal ultrafilter on $\lambda.$