Measure
Definition
A (positive) measure $\mu$ on $X$ is a map $\mu:\mathcal{A} \rightarrow \left. \lbrack 0,\infty\rbrack \right.$ satisfying the following conditions:
(M${}_{0}$): $\mathcal{A}$ is a sigma-algebra in $X$.
(M${}_{1}$): $\mu\left. (\varnothing) \right. = 0$.
(M${}_{2}$): $\left. \left( A_{n} \right) \right._{n \in \mathbb{N}} \subseteq \mathcal{A}\mathrm{\text{ pairwise disjoint }} \Longrightarrow \mu\left. \left( \bigsqcup(n \in \mathbb{N})A_{n} \right) \right. = \sum_{n \in \mathbb{N}}\mu\left. \left( A_{n} \right) \right.$.
where $\sqcup$ represents the union of disjoint sets.