Let \((X, \mcm)\) be a measurable space. A measure on \((X,\mcm)\) is a function \(\mu\colon \mcm \to [0,\infty]\) such that
\(\mu(\emptyset) = 0\), and
\(\mu\) is countably additive: if \((E_k)_{k\geq 1}\) is a sequence of pairwise disjoint sets in \(\mcm\), then \[ \mu\qty{\bigcup_{k\geq 1} E_k} = \sum_{k\geq 1} \mu(E_k). \]
Countable additivity is also called \(\sigma\)-additivity. A function \(\mu\colon\mcm\to[0,\infty]\) with \(\mu(\emptyset)=0\) for which the identity in (2) is required only for finite families of pairwise disjoint sets is called finitely additive.