Lets say you have a probability space $(\Omega, \mathcal{F},P)$ And a stochastic process on this space $\{X_t, t \in T\}.$ Assume that our process takes values in $\mathbb{R}$ and $T$ is a totally ordered set.
You also have a filtration $\{\mathcal{F}_t, t\in T\}$, where for each t, and each open set $U \subset \mathbb{R}$, $X_t^{-1}(U)\in \mathcal{F}_t$. We also have that for $s <t$, $\mathcal{F}_s\subset\mathcal{F}_t\subset\mathcal{F}$.
What I do not understand is how the filtration in some ways model information up to time $t$? Could you please explain this? In mathematical finance, it is said that the filtration models what information the investor has at time t? But I do not see the connection really; how does the filtration really tell us what kind of information we have?