# 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

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

- Results about
**measures**can be found**here**.

## Sources

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