Definition:Random Variable/General Definition/Notation

Notation
As an abuse of notation, we may write:


 * $\set {\omega \in \Omega : \map P {\map X \omega} }$ as $\set {\map P X}$

for any propositional function of $\map X \omega$ such that:


 * $\set {\omega \in \Omega : \map P {\map X \omega} }$ is $\Sigma$-measurable.

We may therefore write, for example:


 * $\map \Pr {\set {\omega \in \Omega : \map X \omega \in B} }$

for some $B \in \Sigma'$, as:


 * $\map \Pr {\set {X \in B} }$

Usually the curly brackets are dropped and we write:


 * $\map \Pr {\set {\omega \in \Omega : \map X \omega \in B} } = \map \Pr {X \in B}$