Definition:Measure (Measure Theory)

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This page is about measures in the context of Measure Theory. For other uses, see Measure.

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.


Definition 1

$\mu$ 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


Definition 2

$\mu$ 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

  • Results about measures can be found here.


Sources