nLab filtered probability space

Idea

In probability theory, a sigma-algebra can be interpreted as a system of possible “distinctions” that can be made in a measurable or probability space $X$.

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:

• $I$ is a partially ordered set (often totally ordered), with finitary upper bounds;
• 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.

Limits and colimits

(…)

Related concepts

• probability space, random variable, stochastic process
• martingale

filtered objects

• filtered topological space

  • spectral sequence of a tower of fibrations

• filtered vector space

• filtered chain complex

  • spectral sequence of a filtered complex

  • Hodge filtration

• filtered ring

• filtered object in an (∞,1)-category

• filtered stable homotopy type

  • spectral sequence of a filtered stable homotopy type

• filtered probability space

associated graded objects

• associated graded vector space

• associated graded ring

References

• Ruben Van Belle, A categorical treatment of the Radon-Nikodym theorem and martingales, 2023. (arXiv)

• Paolo Perrone and Ruben Van Belle, Convergence of martingales via enriched dagger categories, 2024. (arXiv)

Introductory material

• Paolo Perrone, Starting Category Theory, World Scientific, 2024, Section 3.2.7. (website)

• Paolo Perrone, Convergence of martingales via enriched dagger categories, video talk, 2024. (YouTube)

category: probability