Definition:Probability Measure/Definition 3

From ProofWiki
Jump to navigation Jump to search


Let $\EE$ be an experiment.

Let $\Omega$ be the sample space on $\EE$.

Let $\Sigma$ be the event space of $\EE$.

A probability measure on $\EE$ is a mapping $\Pr: \Sigma \to \R$ which fulfils the following axioms:

\((\text I)\)   $:$     \(\ds \forall A \in \Sigma:\)    \(\ds \map \Pr A \)   \(\ds \ge \)   \(\ds 0 \)      
\((\text {II})\)   $:$      \(\ds \map \Pr \Omega \)   \(\ds = \)   \(\ds 1 \)      
\((\text {III})\)   $:$     \(\ds \forall A \in \Sigma:\)    \(\ds \map \Pr A \)   \(\ds = \)   \(\ds \sum_{\bigcup \set e \mathop = A} \map \Pr {\set e} \)      where $e$ denotes the elementary events of $\EE$

Also see