Definition:Measure (Measure Theory)

Definition
Let $\left({X, \mathcal A}\right)$ be a measurable space.

Let $\mu: \mathcal A \to \overline{\R}$ be a 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.

Alternative Definition
Alternatively, condition $(3)$ may be replaced by:


 * $(3'):\quad \mu \left({\varnothing}\right) = 0$

Note that it is assured that $\varnothing \in \mathcal A$ by Sigma-Algebra Contains Empty Set.

That the two definitions are equivalent is shown on Equivalence of Definitions of Measure (Measure Theory).

Elementary Consequences
It follows from Measure of Empty 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.:
 * $\forall A, B \in \mathcal A: A \cap B = \varnothing \implies \mu \left({A \cup B}\right) = \mu \left({A}\right) + \mu \left({B}\right)$

Also see

 * Lebesgue Measure
 * Measure Space


 * Characterization of Measures