Definition 5.1.2 (Measure Space).label Let $(X,\Si)$ be a measurable space and $\mu:\Si\to[0,\infty]$ then $\mu$ is a (countably additive) measure if

1. (M1)

   $\mu(\emp)=0$

2. (M2)

   For any $\curl{E_n}_{n\in\N}\suf\Si$ pairwise disjoint, $\mu(\dcups{}{n\in\N}E_{n})=\sums{}{n\in\N}\mu(E_{n})$

In which case, $(X,\Si,\mu)$ is a measure space. If $\mu$ instead satisfies (M1) and

1. (M2’)

   For any $\curl{E_n}_{1}^{n}\suf\Si$ pairwise disjoint, $\mu(\dcups{n}{i=1}E_{i})=\sums{n}{i=1}\mu(E_{i})$

then $\mu$ is a finitely-additive measure.