Carathéodory criterion

Carathéodory’s criterion is the clever test that decides which sets an outer measure can handle additively. The slogan: a set E is measurable if and only if it splits every other set cleanly, with no slack. That is, using E as a knife to cut any test set must leave the two pieces adding back up to the whole, with respect to the outer measure.

Precisely, given an outer measure mu-star on X, a set E is Carathéodory measurable if for every test set A (every subset of X), mu-star(A) = mu-star(A intersect E) + mu-star(A minus E). Because subadditivity already gives the “at most” direction automatically, the content of the criterion is the reverse inequality: E must not waste any measure when it partitions A.

The remarkable theorem of Carathéodory is that the collection of all such measurable sets is automatically a sigma-algebra, and the outer measure restricted to it is a genuine, complete, countably additive measure. This is the engine that turns the easy-to-build outer measure into a real measure; applied to Lebesgue outer measure it yields exactly the Lebesgue measurable sets and Lebesgue measure.