Definition:Measure (Measure Theory)

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

• Equivalence of Definitions of Measure (Measure Theory)

• Sigma-Algebra Contains Empty Set

• Measure of Empty Set is Zero: $\map \mu \O = 0$.
• Measure is Finitely Additive Function

• Definition:Lebesgue Measure
• Definition:Measure Space

• Characterization of Measures

• Results about measures can be found here.

Sources

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• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $4.1$