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Carathéodory's theorem begins with a set function defined on a ring of sets which is already a pre-measure.

The theorem extends the pre-measure to the sigma-algebra generated by the initial ring.

The measure extended by Carathéodory's theorem is complete: every subset of a null set is measurable.

A set $E$ is measurable with respect to the outer measure iff for all sets $A$, $\mu^*(A) = \mu^*(A \cap E) + \mu^*(A \cap E^c)$, where $\mu^*$ is the outer measure.