Example of filtration

Question

Could someone please provide an example of such filtration:

Answer 1

The sigma-algebras $\mathfrak I_t$ make up the filtration in that link. To get an example, you should take an example of a one-to-one ergodic measure-preserving transformation $\mathbb U$, and a "starting" sigma-algebra $\mathfrak I_0$, and then plug them into the formula.

We say that an event $\Lambda$ is "known" at a future time $t$ iff $\mathbb U^{-t} (\Lambda)$ is known at time $0$.

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An example.
It seems we are working with two-sided sequences. Let $\Omega = \{0,1\}^{\mathbb Z}$ be the space of two-sided sequences of $0,1$. $\mathfrak G$ is the product sigma-algebra (equivalently, the Borel sets for the product topology). Let $\pi_t$ be the projection onto the $t$ coordinate. That is, if $\omega \in \Omega$ is $$ \omega = (\dots,\omega_{-2},\omega_{-1},\omega_0;\omega_1,\omega_2,\cdots), \tag1$$ then $\pi_t(\omega) = \omega_t$. I put a semicolon in $(1)$ to mark where the $0$ coordinate is. We think of $\omega_0$ as what is happening now, $\omega_{-1}$ is what happend yesterday, $\omega_{-2}$ what happened the day before yesterday, and so on. And $\omega_1$ us what will happen tomorrow, $\omega_2$ the day after tomorrow, and so on. The events $\Lambda \in \mathfrak G$ that we know now make up the sigma-algebra $$ \mathfrak I_0 = \sigma\{\pi_t\;:\;t \le 0\} \tag2$$ So saying that $\Lambda \in \mathfrak I_0$ means that it depends only on what happens today, what happened yesterday, and so on backward. These are the events we know now. (Assuming we have kept perfect records of the past, but have no crystal ball to fortell the future.)

Let $\mathbb P : \mathfrak G \to [0,1]$ be the "product" measure on $\Omega$, where each factor has the $(1/2,1/2)$ measure. In probabilistic language, the random variables $\pi_t$ are i.i.d Bernoulli trials. The transformation $\mathbb U: \Omega \to \Omega$ moves everything one step into the past. $$ \mathbb U\big( (\dots,\omega_{-2},\omega_{-1},\omega_0;\omega_1,\omega_2,\cdots) \big) = (\dots,\omega_{-1},\omega_0,\omega_1;\omega_2,\omega_3,\cdots). $$ Equivalently, $\pi_{t}(\mathbb U(\omega)) = \pi_{t-1}(\omega)$. (If there is an event that the Tomorrow People think will happen three days in their future, we think it will happen four days in our future.) I leave it to you to check that it is an ergodic measure-preserving transformation. So $$ \mathfrak I_1 = \{\Lambda \in \mathfrak G \;:\; \mathbb U^{-1}\Lambda \in \mathfrak I_0\} = \sigma\{\pi_t\;:\;t \le 1\} \tag3$$ That is, saying $\Lambda \in \mathfrak I_1$ means that we will know tomorrow whether $\Lambda$ occurs or not. Similarly, $$ \mathfrak I_s = \sigma\{\pi_t\;:\;t \le s\} $$ and saying $\Lambda \in \mathfrak I_s$ means that we know on day $s$ whether $\Lambda$ occurs or not. (Here $s \in \mathbb Z$. It could be negative.)