Definition:Measure (Measure Theory)

Definition
Let $\mathcal A$ be a $\sigma$-algebra.

Let $\mu: \mathcal A \to \overline {\R}$ be a real-valued function where $\overline {\R}$ denotes the set of extended real numbers.

Then $\mu$ is called a measure on $\mathcal A$ iff it has the following properties:

$(1) \quad$: For every $S \in \mathcal A$:
 * $\mu \left({S}\right) \ge 0$

$(2) \quad$: For every sequence of pairwise disjoint sets $\left\{{S_{n}}\right\} \subseteq \mathcal A$:
 * $\displaystyle \mu \left({\bigcup_{n=1}^{\infty} S_n}\right) = \sum_{n=1}^{\infty} \mu \left({S_{n}}\right)$

(that is, $\mu\ $ is a countably additive function).

$(3) \quad$: There exists at least one $A \in \mathcal A$ such that $\mu \left({A}\right)$ is finite.

Elementary Consequences
It follows from Measure of Null Set is Zero that $\mu \left({\varnothing}\right) = 0$.

It then follows from Measure is Finitely Additive Function that $\mu$ is also finitely additive, i.e.:
 * $\forall A, B \in \mathcal A: A \cap B = \varnothing \implies \mu \left({A \cup B}\right) = \mu \left({A}\right) + \mu \left({B}\right)$

Note
The definition of a measure is usually given either that:


 * The measure is defined on $\mu: \mathcal A \to \R$, i.e. that the domain of $\mu$ consists entirely of finite numbers;


 * The measure of the empty set is defined as being zero.

In either case, it can be seen from the above that both of these conditions follow from the definition as given on this page.