Premeasure

A premeasure is a countably additive nonnegative set function on an algebra of sets, with countable additivity required whenever the union remains in that algebra.

A normalized positive linear functional on $C(\Omega )$ for Cantor space gives a finitely additive set function on the algebra of clopen sets. If a countable disjoint union of such sets is itself clopen, compactness gives a finite subcover; all other terms are empty. Finite additivity therefore gives the required countable additivity, making $\ell$ a premeasure. The Caratheodory extension theorem produces a unique Borel probability measure.

A premeasure on an algebra of sets extends to a measure on the generated sigma-algebra. The extension is unique when the premeasure is sigma-finite.