Category:Definitions/Signed Measures

From ProofWiki
Jump to navigation Jump to search

This category contains definitions related to Signed Measures.
Related results can be found in Category: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