# proof about a sigma algebra generated by a random variable

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 $$\sigma(Y^{-1}(\mathcal{B})) \subset \sigma(Y)$$.
• Hello, when you say "obviously Y is $\sigma(Y^{-1} (B))$ measurable", could you explain why? Oct 25, 2019 at 3:15
• $\forall B \in \mathcal{B}$, $\{Y \in B\} \in Y^{-1}(\mathcal{B}) \subset \sigma(Y^{-1}(\mathcal{B}))$, so $Y$ is $\sigma(Y^{-1}(\mathcal{B}))$-measurable. Oct 25, 2019 at 3:17
• could you explain why "$\sigma(Y)$" is the smallest sigma algebra that makes $Y$ measurable? Oct 25, 2019 at 3:21
• That's exactly how "$\sigma$-algebra generated by a random variable" defines. See Probability: Theory and Examples Version 5 by Rick Durrett, page 15, Section 1.3 Random Variable Oct 25, 2019 at 3:24
• Hello, yes it says $\sigma(Y)$ is the smallest sigma algebra generated by $Y$. They always denote $\sigma(Y)$ as "smallest". But is it possible to know why it is the smallest? how do you know it is the smallest? Oct 25, 2019 at 3:35