# 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 $\Sigma$ if and only if $\mu$ has the following properties:

 $(1)$ $:$ $\displaystyle \forall E \in \Sigma:$ $\displaystyle \map \mu E$ $\displaystyle \ge$ $\displaystyle 0$ $(2)$ $:$ $\displaystyle \forall \sequence {S_n}_{n \mathop \in \N} \subseteq \Sigma: \forall i, j \in \N: S_i \cap S_j = \O:$ $\displaystyle \map \mu {\bigcup_{n \mathop = 1}^\infty S_n}$ $\displaystyle =$ $\displaystyle \sum_{n \mathop = 1}^\infty \map \mu {S_n}$ that is, $\mu$ is a countably additive function $(3)$ $:$ $\displaystyle \exists E \in \Sigma:$ $\displaystyle \map \mu E$ $\displaystyle \in$ $\displaystyle \R$ that is, there exists at least one $E \in \Sigma$ such that $\map \mu E$ is finite

## 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:

$\forall E, F \in \Sigma: E \cap F = \O \implies \map \mu {E \cup F} = \map \mu E + \map \mu F$