Definition:Random Variable/Real-Valued/Notation

Notation
As an abuse of notation, we may write:


 * $\set {\omega \in \Omega : \map X \omega \le x}$ as $\set {X \le x}$
 * $\set {\omega \in \Omega : \map X \omega \ge x}$ as $\set {X \ge x}$
 * $\set {\omega \in \Omega : \map X \omega < x}$ as $\set {X < x}$
 * $\set {\omega \in \Omega : \map X \omega > x}$ as $\set {X > x}$
 * $\set {\omega \in \Omega : \map X \omega = x}$ as $\set {X = x}$
 * $\set {\omega \in \Omega : \map X \omega \in A}$ as $\set {X \in A}$

Generally, we 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:


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

as:


 * $\map \Pr {\set {X = x} }$

Usually the curly brackets are dropped and we write:


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