I need some practice finding the $\sigma$-algebra generted by a radnom variable, but I can't find any worked examples online. I was wondering if anyone could work through the solution to the following example I've made up:
Let $X : ([0,1],\mathbb{P}) \to (\mathbb{R}, \mathscr{B})$, where $\mathbb{P} = \lambda$ is the lebesgue measure ($\lambda[a,b] = b-a$) and $\mathscr{B}$ is the Borel sigma algebra. Define: $$X(\omega) = \begin{cases} 0 &x <1/3\\ 1 & x \in [1/3, 2/3]\\ 2x & x \in (2/3, 1]\\ \end{cases} $$
Intuitively, I think it might be something like $$\sigma(X) = \{X^{-1}(B): B \in \mathscr{B}\} = \sigma(\{[0,1/3),[1/3,2/3],\mathscr{B}((2/3,1]),[0,1]\})$$ Am I right, and how do I make this rigorous? If anyone could point me to more worked examples that would also be good. Thanks!