Given a set X, a set function μ^* :2^X->[0, ∞] is said to be an outer measure provided that μ^*(∅) = 0 and that μ^* is countably monotone, where ∅ is the empty set. Given a collection S of subsets of X and an arbitrary set function μ:S->[0∞], one can define a new set function μ^* by setting μ^*(∅) = 0 and defining, for each non-empty subset E subset X, μ^*(E) = inf sum_(k = 1)^∞ μ(E_k) where the infimum is taken over all countable collections {E_k}_(k = 1)^∞ of sets in S which cover E. The resulting function μ^* :2^X->[0∞] is an outer measure and is called the outer measure induced by μ.

Carathéodory measure | countable monotonicity | measurable set | measurable space | measure | measure space | premeasure