An outer measure on a set X is a map m* from the power set of X to [0,∞] such that

• m*(⌀) = 0;

• m* is monotonic: if A ⊆ B, then m*(A) ≤ m*(B);

• m* is countably subadditive: $m^{*}\left(\bigcup _1^{\infty }{A}_k\right)\le \sum _1^{\infty }{m}^{*}\left(A_k\right)$.

Given an outer measure m*, a set E is called Carathéodory measurable if

for all subsets A of X. The Carathéodory measurable sets form a sigma algebra on which m* is a measure.