Definition 14.7.5 (Premeasure). Let $X$ be a set, $\alg \subset 2^{X}$ be a ring, and $\mu_{0}: \alg \to [0, \infty]$, then $\mu_{0}$ is a premeasure if:

1. $\mu_{0}(\emptyset) = 0$.

2. For any $\seq{A_n}\subset \alg$ pairwise disjoint with $\bigsqcup_{n \in \natp}A_{n} \in \alg$,

   \[\mu_{0}\paren{\bigsqcup_{n \in \natp}A_n}= \sum_{n \in\natp}\mu_{0}(A_{n})\]