Definition:Complex Measure

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Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu : \Sigma \to \C$ be a function.

We say that $\mu$ is a complex measure on $\struct {X, \Sigma}$ if and only if:

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

Also see

  • Results about complex measures can be found here.