Definition:Concentration on Measurable Set
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Definition
Let $\struct {X, \Sigma}$ be a measurable space.
Measure
Let $\mu$ be a measure on $\struct {X, \Sigma}$.
Let $E \in \Sigma$.
We say that $\mu$ is concentrated on $E$ if and only if:
- $\map \mu {E^c} = 0$
Signed Measure
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $\size \mu$ be the variation of $\mu$.
Let $E \in \Sigma$.
We say that $\mu$ is concentrated on $E$ if and only if:
- $\map {\size \mu} {E^c} = 0$
Complex Measure
Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.
Let $\size \mu$ be the variation of $\mu$.
Let $E \in \Sigma$.
We say that $\mu$ is concentrated on $E$ if and only if:
- $\map {\size \mu} {E^c} = 0$