Axiom:Measure Axioms/Formulation 2

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 $\mu$ 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 \map \mu \O $	$\ds = $	$\ds 0 $

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

Axiom:Measure Axioms/Formulation 2

Definition

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