Premeasure

A premeasure on a set algebra $\mathcal A$ is a function $\mu_0:\mathcal A\to[0,\infty]$ with $\mu_0(\varnothing)=0$ such that whenever $(A_n)_{n\ge 1}$ is a pairwise disjoint sequence in $\mathcal A$ whose union $\bigcup_{n=1}^\infty A_n$ also lies in $\mathcal A$, one has

Here $\varnothing$ is the empty set .

Premeasures are typically defined on a set algebra that is simpler than a full sigma-algebra, and then extended to a measure using the Carathéodory construction .

Examples:

• On the set algebra of finite unions of half-open intervals $[a,b)\subseteq\mathbb R$, define $\mu_0([a,b))=b-a$ and extend additively across disjoint unions; this is a premeasure.
• On a set $X$, let $\mathcal A$ be the set algebra of finite subsets of $X$ and define $\mu_0(A)=|A|$ (cardinality) for $A\in\mathcal A$; this is a premeasure (the “counting premeasure” on finite sets).