Premeasure - Knowlpedia

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

\mu_0\!\left(\bigcup_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty \mu_0(A_n).

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 $[0,1]$ , let $\mathcal A$ be the set algebra generated by half-open intervals. Assigning each interval its length and extending additively over finite disjoint unions defines a premeasure on $\mathcal A$ .
On any set $X$ , the counting function $\mu_0(A)=|A|$ , with value $+\infty$ for infinite $A$ , is a premeasure on the power set $\mathcal P(X)$ .