Sigma-Field Generated by A Random Variable

Definition

Suppose $(\Omega ,\mathcal{F},P)$ be a Probability Space and a Random Variable $Y:\Omega \to \mathbb{R}$. The Sigma-Field generated by $Y$ is $\sigma (Y)=\{Y^{-1}(B)|B\in \mathcal{R}\}\subset \mathcal{F}$ where $\mathcal{R}$ is Borel Sigma-Field