Definition:Complex Measure
Jump to navigation
Jump to search
Definition
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.
Sources
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.1$