linguistic:paraphrasing, structural:expansion
A note on notation

Even though events are subsets of some sample space Ω, {\displaystyle \Omega,} they are often written as predicates or indicators involving random variables. For example, if X {\displaystyle X} is a real-valued random variable defined on the sample space Ω, {\displaystyle \Omega,} the event { ω ∈ Ω ∣ u < X (ω) ≤ v } {\displaystyle \{\omega \in \Omega \mid u<X(\omega)\leq v\}\,} can be written more conveniently as, simply, u < X ≤ v. {\displaystyle u<X\leq v\,.} This is especially common in formulas for a probability, such as Pr (u < X ≤ v) = F (v) − F (u). {\displaystyle \Pr(u<X\leq v)=F(v)-F(u)\,.} The set u < X ≤ v {\displaystyle u<X\leq v} is an example of an inverse image under the mapping X {\displaystyle X} because ω ∈ X − 1 ((u, v ]) {\displaystyle \omega \in X^{-1}((u,v])} if and only if u < X (ω) ≤ v. {\displaystyle u<X(\omega)\leq v.}