In my classes on stochastic processes we often refer to $\sigma$-Algebras as information or as the history or future of some process $X$.
E.g.
$$ \mathcal{F}^{X}_{[0, t]} = \sigma(X_s , 0 \leq s \leq t)$$
we described as the history up to point $t \in T$ where $T \subset [0, \infty)$.
Generally speaking I understand that a $\sigma$-Algebra is a set of sets with specific properties (like being closed under countable unions etc) and that an element of a $\sigma$-Algebra is interpreted as an event that can occur when conducting a random experiment.
What I don't understand is how we can regard this as information or history since the $\sigma$-Algebra doesn't say what happens or happened but only which events could occur or could have occurred. After all, for every event $A$ in the $\sigma$-Algebra the complement $A^C$ is also contained.
