Premeasure

A premeasure on an algebra of sets $\mathcal{A}$ is a function ${\mu }_0:\mathcal{A}\to [0,\infty ]$ with ${\mu }_0(\varnothing )=0$ and countable additivity for every disjoint sequence whose union remains in $\mathcal{A}$. The restriction on the union is necessary because an algebra of sets need not be a sigma-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.