# Category:Signed Measures

This category contains results about Signed Measures.

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 $\struct {X, \Sigma}$ 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

## Subcategories

This category has the following 12 subcategories, out of 12 total.

## Pages in category "Signed Measures"

The following 17 pages are in this category, out of 17 total.