I am going through Klenke's Probability Theory. Here are the definitions he presents for adapted
Definition 9.10 A stochastic process $X = (X_t,t \in I)$ is called adapted to the filtration $\mathbb{F}$ if $X_t$ is $\mathcal{F}_t$-measurable for all $t \in I$.
and for predictable
Definition 9.12 A stochastic process $X = (X_n,n\in \mathbb{N}_0)$ is called predictable with respect to the filtration $\mathbb{F} = (\mathcal{F}_n,n \in \mathbb{N}_0)$ if $X_0$ is constant and if, for every $n \in \mathbb{N}$ $X_n$ is $\mathcal{F}_{n-1}$-measurable.
The definition of predictable makes sense to me if $\mathbb{F} = \sigma(X)$, but when this is not the case, I do not know how to predict future values in $X_n$.
I constructed a concrete example. Let $(\Omega,\mathcal{F},\mathbb{P}) = ([0,1),\mathcal{B}(\Omega),\mu)$. Then, take the following process. Let $X_0(\omega) = 0$. Then, for $i \geq 1$, let $\omega_j$ be the jth digit in the decimal expansion of $\omega$ and $$X_i(\omega) = \begin{cases} 1 & \omega_j \text{ even} \\ 0 & \omega_j \text{ odd}.\end{cases}$$
If we let $\mathbb{F} = \mathcal{F}$ (giving out the original $\sigma$-algebra at each time step), then each $X_i$ is $\mathcal{F}_0$-measurable. Now we start the process and find that $X_0 =0$. How do I then predict the value of $X_1$?