Definition:Concentration on Measurable Set/Complex Measure/Definition 2

Definition

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

Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.

Let $E \in \Sigma$.

We say that $\mu$ is concentrated on $E$ if and only if:

for every $\Sigma$-measurable set $A \subseteq E^c$, we have $\map \mu A = 0$.

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.3$: Singularity