I am studying Martingales and Stopping Times from the 3rd edition of the book "Probability and Measure" by Patrick Billingsley. The following arose while I was reading page 465.
Let $(\varOmega,\mathscr{F},\mathsf{P})$ be a probability space, let $(X_n)_{n\in\mathbb{N}}$ be a sequence of random variables on $(\varOmega,\mathscr{F})$, and set $(\forall n\in\mathbb{N})\;\mathscr{F}_n=\sigma(X_i)_{0\leq i\leq n}$. Suppose that $(X_n)_{n\in\mathbb{N}}$ is a martingalge relative to the filtration $(\mathscr{F}_n)_{n\in\mathbb{N}}$ and let $\tau\colon\varOmega\to\mathbb{N}$ be a stopping time relative to $(\mathscr{F}_n)_{n\in\mathbb{N}}$. Set $$\mathscr{F}_\tau=\left\{A\in\mathscr{F}\;\mid\; (\forall n\in\mathbb{N})\;A\cap\left[\tau=n\right]\in\mathscr{F}_n\right\}, $$ where $\left[\tau=n\right]=\left\{\omega\in\varOmega\;\mid\;\tau(\omega)=n\right\}$.
I am interested in finding a subset $\mathcal{A}$ of $\mathscr{F}_\tau$ such that $\sigma(\mathcal{A})=\mathscr{F}_\tau$. In particular, since it is easy to show that $$(\forall n\in\mathbb{N})\quad \mathscr{F}_n=\sigma\Bigg(\left\{\bigcap_{i=0}^nX_i^{-1}(B_i)\;\mid\; (B_i)_{0\leq i\leq n}\text{ are Borel sets in }\mathbb{R}\right\}\Bigg), $$ I guess: if we set $$ \mathcal{A}=\left\{\left[\tau=n\right]\cap\bigcap_{i=0}^nX_i^{-1}(B_i)\;\mid\; n\in\mathbb{N}\text{ and }(B_i)_{0\leq i\leq n}\text{ are Borel sets in }\mathbb{R}\right\}, $$ then $\sigma(\mathcal{A})=\mathscr{F}_\tau$. However, I have no clue whether this is true or not. All I could check is that $\mathcal{A}\subset\mathscr{F}_\tau$.
My question: Is my guess correct? If so, how could I establish that $\sigma(\mathcal{A})=\mathscr{F}_\tau$, where $\mathcal{A}$ is defined above?
Any help/hint is highly appreciated.
