# Definition:Measure (Measure Theory)

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## 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.

Then $\mu$ is called a **measure** on $\Sigma$ if and only if $\mu$ has the following properties:

\((1)\) | $:$ | \(\displaystyle \forall E \in \Sigma:\) | \(\displaystyle \map \mu E \) | \(\displaystyle \ge \) | \(\displaystyle 0 \) | |||

\((2)\) | $:$ | \(\displaystyle \forall \sequence {S_n}_{n \mathop \in \N} \subseteq \Sigma: \forall i, j \in \N: S_i \cap S_j = \O:\) | \(\displaystyle \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} \) | \(\displaystyle = \) | \(\displaystyle \sum_{n \mathop = 1}^\infty \map \mu {S_n} \) | that is, $\mu$ is a countably additive function | ||

\((3)\) | $:$ | \(\displaystyle \exists E \in \Sigma:\) | \(\displaystyle \map \mu E \) | \(\displaystyle \in \) | \(\displaystyle \R \) | that is, there exists at least one $E \in \Sigma$ such that $\map \mu E$ is finite |

## Alternative Definition

Alternatively, condition $(3)$ may be replaced by:

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

Note that it is assured that $\O \in \Sigma$ by Sigma-Algebra Contains Empty Set.

That the two definitions are equivalent is shown on Equivalence of Definitions of Measure (Measure Theory).

## Elementary Consequences

It follows from Measure of Empty Set is Zero that $\map \mu \O = 0$.

It then follows from Measure is Finitely Additive Function that $\mu$ is also finitely additive, that is:

- $\forall E, F \in \Sigma: E \cap F = \O \implies \map \mu {E \cup F} = \map \mu E + \map \mu F$

## Also see

## Sources

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