I have a question.
Let Y be a real-valued random variable defined on a probability space ($\Omega$, $F$, $P$) where Y:$\Omega \longrightarrow R$.
Show that the $\sigma$-algebra $\sigma(Y)$ generated by the random variable $Y$ coincides with the $\sigma$-algebra $\sigma(Y^{-1}{(\mathcal{B}}))$ generated by the collection of events $Y^{-1}(\mathcal{B})$= { $ Y^{-1}(B) | B \in \mathcal{B} $ }. The $\mathcal{B}$ is the Borel $\sigma$-algebra.
I don't have much clue about how to approach this question. Could someone comments?
It is just a self study problem that I have.
Just a game of logic. Use the minimality of the two $\sigma$-algebra's.
Obviously $Y$ is $\sigma(Y^{-1}(\mathcal{B}))$-measurable, so we must have $\sigma(Y) \subset \sigma(Y^{-1}(\mathcal{B}))$ because $\sigma(Y)$ is the smallest $\sigma$-algebra makes $Y$ measurable. For the otehr direction, $\forall A= \{Y\in B\} \in Y^{-1}(\mathcal{B})$, where $B$ is an arbitrary element of $Y^{-1}(\mathcal{B})$, we have $ A \in \sigma(Y)$ by the definition of $\sigma(Y)$. Since $\sigma(Y^{-1}(\mathcal{B}))$ is the smallest $\sigma$-algeba containing $Y^{-1}(\mathcal{B})$, we also have $\sigma(Y^{-1}(\mathcal{B})) \subset \sigma(Y)$.