Definition:Signed Measure

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

Let $\mu : \Sigma \to \overline \R$ be an extended real-valued function such that:


 * if $\map \mu A = +\infty$ for some $A \in \Sigma$, then $\map \mu B > -\infty$ for all $B \in \Sigma$.

and:


 * if $\map \mu A = -\infty$ for some $A \in \Sigma$, then $\map \mu B < +\infty$ for all $B \in \Sigma$.

We say that $\mu$ is a signed measure on $\struct {X, \Sigma}$ :

Also see

 * Definition:Measure
 * Definition:Complex Measure