Definition:Measure (Measure Theory)

Definition
Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu: \Sigma \to \overline \R$ be a mapping, where $\overline \R$ denotes the set of extended real numbers.

Then $\mu$ is called a measure on $\Sigma$ $\mu$ has the following properties:

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


 * $(3'):\quad \map \mu \O = 0$

Note that it is assured that $\O \in \Sigma$ 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 $\map \mu \O = 0$.

It then follows from Measure is Finitely Additive Function that $\mu$ is also finitely additive, that is:
 * $\forall E, F \in \Sigma: E \cap F = \O \implies \map \mu {E \cup F} = \map \mu E + \map \mu F$

Also see

 * Definition:Lebesgue Measure
 * Definition:Measure Space


 * Characterization of Measures