I am wondering what is the interpretation of a sigma-algebra that is not generated by a random variable?
For example if we have a random variable $X$ on a probability space $(\Omega, \mathcal{A},P)$ then the sigma-algebra generated by $X$ is $\sigma(X)=\{X^{-1}(B): B\in \mathcal{B}(\mathbb{R})\}$. Then the interpratation is easy: If we observe what the value of $X$ is, we can know that an event in $\sigma(X)$ has happened or not. Because we know what the value of $X$ is, and then we know if $X\in B$.
However, lets say you have a filtered probability space $(\Omega,\mathcal{F}, (\mathcal{F}_t))_{t \in [0,\infty)},P)$. Often we do not specify that the filtrations are generated by a stochastic process(but we may have a stochastic processes $\{X_t\}_{t \in [0,\infty)}$ adapted to this filtration, and we may have that $\mathcal{F}_t$ is strictly bigger than $\sigma\{\sigma(X_s): s\in [0,t] \}$). In this case, how does then the sigma-algebra $\mathcal{F}_t$"represent all the information up to time t"? Can we say that in this case that at time $t$ we know if the events in $\mathcal{F}_t$ has happened or not? Is this the correct interpretation for an abstract filtration? So at time $t$ I do not know the real $\omega$, but I know if $\omega\in F$ for all $F \in \mathcal{F}_t$?
I guess interpretations are not defined in mathematics. But is this the correct interpretation for abstract sigma-algebras not generated by measurable functions/random variables?