Category:Measures

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This category contains results about measures.

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:

\((1)\)   $:$     \(\ds \forall E \in \Sigma:\)    \(\ds \map \mu E \)   \(\ds \ge \)   \(\ds 0 \)             
\((2)\)   $:$     \(\ds \forall \sequence {S_n}_{n \mathop \in \N} \subseteq \Sigma: \forall i, j \in \N: S_i \cap S_j = \O:\)    \(\ds \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} \)   \(\ds = \)   \(\ds \sum_{n \mathop = 1}^\infty \map \mu {S_n} \)             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