Definition:Measure (Measure Theory)/Definition 1

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.

$\mu$ is called a measure on $\Sigma$ if and only if $\mu$ fulfils the following axioms:

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

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