Definition 5.5.5 (Premeasure).label Let $X$ be a set, $\cali{A}\suf 2^{X}$ be a ring, and $\mu_{0}:\cali{A}\to[0,\infty]$, then $\mu_{0}$ is a premeasure if:

1. (M1)

   $\mu_{0}(\emp)=0$.

2. (PM)

   For any $\curl{A_n}_{n\in\N}\suf\cali{A}$ pairwise disjoint with $\dcups{}{n\in\N}A_{n}\in\cali{A}$,

   \begin{align*}\mu_{0}\parens{\dcups{}{n\in\N}A_n}=\sums{}{n\in\N}\mu_{0}(A_{n})\end{align*}