Complete ultrafilters and measurable cardinals

For an infinite cardinal kappa, a proper filter U on I is kappa-complete if ${\bigcap }_{\xi <\eta }{A}_{\xi }\in U$ whenever $\eta <\kappa$ and each $A_{\xi }\in U$. The intersection with no factors is I. Countably complete means closed under countable intersections, equivalently omega_1-complete in ZFC. Ultrafilter and nonprincipal use Ultrafilter; cardinals use Cardinal (initial ordinal) and cardinality.

A measurable cardinal is an uncountable kappa carrying a nonprincipal kappa-complete ultrafilter on the full power set of kappa. Such a U is a normal measure if every function $f:S\to \kappa$ with $S\in U$, $0\mathrm{\notin }S$ and $f(\alpha )<\alpha$ is constant on a set in U contained in S. A set in U is called measure one.

The associated zero-one set function is $m_U(A)=1$ if $A\in U$, and zero otherwise. It is not the definition of a real-valued measurable cardinal. The fixed-index filter and normality conditions are formulas of ZF and do not themselves use Choice. General cardinality language and the large-cardinal implications on this page use ZFC, with The Axiom of Choice explicit. No ultrafilter or measurable-cardinal existence is asserted.