Definition:Random Variable/General Definition/Notation

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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:

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