outer measure

An outer measure is a first, generous estimate of size that is defined on absolutely every subset, with no admission test required. The idea is to cover a set from outside by simple pieces and take the cheapest such cover; because covering is always possible, every set gets a number. The price of this universality is that an outer measure need not be additive — it only never undershoots.

Formally, an outer measure on a set X is a function mu-star from the power set of X to [0, +infinity] satisfying three conditions: mu-star(empty set) = 0; monotonicity, meaning A contained in B implies mu-star(A) is at most mu-star(B); and countable subadditivity, meaning mu-star of a countable union is at most the sum of the mu-star values of the pieces, even when those pieces overlap or are disjoint.

Outer measures are the standard starting point for constructing genuine measures: one builds an outer measure first (it is easy, requiring only covers), then carves out the sets on which it behaves additively. Carathéodory’s criterion selects exactly those well-behaved sets, and on them the outer measure restricts to a bona fide countably additive measure. So an outer measure is the raw material, and the measure is the refined product.