Let $(E, \mathcal{B}(E))$ be a measurable space. A measure on $(E,\mathcal{B}(E))$ is a function $\mu\colon \mathcal{B}(E) \to \mathbb{R} \union \{\infty\}$ with values in the extended real numbers such that:

1. $\mu(A) \geq 0$ for $A \in \mathcal{B}(E)$ , with equality if $A = \emptyset$
2. $\mu(\bigcup_{i=0}^\infty A_i) = \sum_{i=0}^\infty \mu(A_i)$ for any sequence of pairwise disjoint sets $A_i \in \mathcal{B}(E)$ .

Occasionally, the term positive measure is used to distinguish measures as defined here from more general notions of measure which are not necessarily restricted to the non-negative extended reals.

The second property above is called countable additivity, or $\sigma$ -additivity. A finitely additive measure $\mu$ has the same definition except that $\mathcal{B}(E)$ is only required to be an algebra and the second property above is only required to hold for finite unions. Note the slight abuse of terminology: a finitely additive measure is not necessarily a measure.

The triple $(E, \mathcal{B}(E), \mu)$ is called a measure space. If $\mu(E) = 1$ , then it is called a probability space, and the measure $\mu$ is called a probability measure.

Lebesgue measure on $\mathbb{R}^n$ is one important example of a measure.