information:extraction, linguistic:voice_change
Real-valued random variables

In this case the observation space is the set of real numbers. Recall, (Ω, F, P) {\displaystyle (\Omega,{\mathcal {F}},P)} is the probability space. For a real observation space, the function X: Ω → R {\displaystyle X\colon \Omega \rightarrow \mathbb {R} } is a real-valued random variable if

This definition is a special case of the above because the set { (− ∞, r ]: r ∈ R } {\displaystyle \{(-\infty,r]:r\in \mathbb {R} \}} generates the Borel σ-algebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that { ω: X (ω) ≤ r } = X − 1 ((− ∞, r ]) {\displaystyle \{\omega:X(\omega)\leq r\}=X^{-1}((-\infty,r])}.