# Definition:Measure (Measure Theory)

## Definition

Let $\left({X, \Sigma}\right)$ 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 $\Sigma$ iff it has the following properties:

- $(1): \quad$ For every $E \in \Sigma$:
- $\mu \left({E}\right) \ge 0$

- $(2): \quad$ For every sequence of pairwise disjoint sets $\left\{{S_{n}}\right\} \subseteq \Sigma$:
- $\displaystyle \mu \left({\bigcup_{n \mathop = 1}^{\infty} S_n}\right) = \sum_{n \mathop = 1}^{\infty} \mu \left({S_{n}}\right)$

- (that is, $\mu\ $ is a countably additive function).

- $(3): \quad$ There exists at least one $E \in \Sigma$ such that $\mu \left({E}\right)$ is finite.

## Alternative Definition

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

- $(3'):\quad \mu \left({\varnothing}\right) = 0$

Note that it is assured that $\varnothing \in \Sigma$ 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 $\mu \left({\varnothing}\right) = 0$.

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

- $\forall E, F \in \Sigma: E \cap F = \varnothing \implies \mu \left({E \cup F}\right) = \mu \left({E}\right) + \mu \left({F}\right)$

## Also see

## Sources

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