A filtration on a probability space is an increasing system of sigma-algebras, for example indexed by time, which one can interpret as “being able to make more and more distinctions”, or “learning more and more as time progresses”.

As the intuition may suggest, it is an instantiation of the categorical notion of filtration, in the context of probability theory.

Definition

A filtered probability space consists of a probability space$(X,\mathcal{A},p)$, equipped with a filtration of sub-sigma-algebras of $\mathcal{A}$, i.e. a collection $(\mathcal{F}_i)_{i\in I}$ where:

For each $i\in I$, $\mathcal{F}_i$ is a sub-sigma-algebra of $\mathcal{A}$;

For each $i\le j\in I$, $\mathcal{F}_i\subseteq \mathcal{F}_j$.

Note that a filtration $(\mathcal{F}_i)_{i\in I}$ as defined above is a filtered object in the category of sigma-algebras, but a cofiltered object? in the category of measurable or probability spaces. Indeed, if $\mathcal{F}_i\subseteq \mathcal{F}_j$, the set-theoretic identity map $X\to X$ is measurable as a function $(X,\mathcal{F}_j)\to(X,\mathcal{F}_i)$, but not the other way around.