Definition:Measure (Measure Theory)/Definition 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.

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

Also see

• Equivalence of Definitions of Measure (Measure Theory)

• Results about measures can be found here.

Sources

This page may be the result of a refactoring operation. As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering. If you have access to any of these works, then you are invited to review this list, and make any necessary corrections. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{SourceReview}} from the code.

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): measure (S)