Information theory and Shannon's formula provide a measure of uncertainty in specific situations where the outcomes and their probabilities are known. But in many situations, this information theory does not provide an adequate answer for the amount of relevant information (e.g., a string of n bits maximizes Shannon entropy if it is random: we have the maximum uncertainty, but the string probably does not contain much relevant information).
On the other hand, in the theory of stochastic processes, there is an adapted filtration $\mathcal{F}_t$ that "accounts" for the available information up to a certain instant. For each time $t$, $\mathcal{F}_t$ is a $\sigma$-algebra, but I wonder if from it we could construct a quantitative measure of the relevant iformation up to time $t$.