# Definition:Measure (Measure Theory)

## 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.