Definition 18.1.2 (Measure).label Let $(X, \cm)$ be a measurable space and $\mu: \cm \to [0, \infty]$, then $\mu$ is a (countably-additive) measure if

1. (M1)

   $\mu(\emptyset) = 0$.

2. (M2)

   For any $\seq{E_n}\subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{n \in \natp}E_n}= \sum_{n \in \natp}\mu(E_{n})$.

In which case, $(X, \cm, \mu)$ is a measure space.

If $\mu: \cm \to [0, \infty]$ instead satisfies (M1) and

1. (M2’)

   For any $\seqf{E_j}\subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{j = 1}^n E_j}= \sum_{j = 1}^{n} \mu(E_{j})$.

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