Outer measures and the Carathéodory construction

An outer measure assigns zero to the empty set, respects inclusion, and is countably subadditive. The last property says covering a union never costs more than covering its pieces separately. Equality for disjoint sets is not required. For Lebesgue outer measure on the line, cover a set by countably many open intervals and take the infimum of their total lengths. A cover need not attain that infimum. Proofs therefore choose covers within an arbitrary positive error and let that error tend to zero.

The use of countably many intervals makes scattered sets inexpensive. For an enumerated set, cover its nth point with an interval of length at most epsilon times two to the minus n. The combined cost is at most epsilon, however the points are distributed. Since epsilon was arbitrary, every countable subset of the line has outer measure zero. In contrast, covering an interval cannot reduce its outer measure below its length; establishing that lower bound uses the geometry of interval covers rather than the definition alone.

Carathéodory calls a set E measurable for an outer measure when every test subset T splits additively into its part inside E and its part outside E. The test set itself need not be measurable. Subadditivity already supplies one inequality, so the real content is the reverse inequality. It forbids an apparent increase in cost merely because the cover must respect the cut. For the outer measure obtained from interval length, intervals satisfy this criterion, giving a starting family of measurable sets.

The remarkable theorem is that all such compatible cuts form a sigma algebra and that the outer measure restricted to it is countably additive. This resulting measure is complete. Indeed, if Z has outer measure zero, every subset of Z has outer measure zero by monotonicity and satisfies the splitting criterion. Completeness is therefore built into this construction. It is not an assertion that every subset of the original space passes the criterion; positive outer measure alone does not establish measurability.

The same strategy works beyond intervals. Begin with a premeasure on an algebra: it is countably additive whenever disjoint algebra sets have a union still belonging to that algebra. Cover arbitrary sets by algebra sets and minimize the sum of their premeasure values. The induced outer measure makes the original algebra measurable and extends its premeasure. Restricting the result to the generated sigma algebra gives the extension theorem used to construct product measures and distributions from simpler set data.

Existence and uniqueness are different conclusions. If the premeasure is sigma-finite, meaning the whole space has a countable cover by algebra sets of finite premeasure, the extension is unique on the generated sigma algebra. Without that hypothesis, uniqueness cannot simply be quoted in this form. The complete Carathéodory domain may also be larger than the generated sigma algebra. Thus a careful construction names the starting algebra, the covering cost, the resulting measurable domain, and the exact domain on which uniqueness is asserted.