Axiom:Measure Axioms/Formulation 3

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
\((3' ')\)	$:$			\(\ds S_i, S_j \in \Sigma, S_i \cap S_j = \O:\)		\(\ds \map \mu {S_i \cup S_j} \)	\(\ds = \)	\(\ds \map \mu {S_i} + \map \mu {S_j} \)

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): measure
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): measure