# Finite filtration

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A finite-time filtration is a sequence of sigma-algebras on the same probability space whose information increases with time.

mko-finite-filtration · definition · v0.10.0



## Mathematical statement

Represent each time's sigma-algebra by a partition of the same finite sample space. F_s being contained in F_t means the later partition refines the earlier one. A filtration specifies information nesting only: a process is not automatically adapted, and no natural-filtration, independence or Markov claim follows.





## Representative formula (exact TeX)

F(t)





## Assumptions


- Every partition uses the same outcomes 0 through n−1; labels within a row indicate only block equivalence.
- Refinement is checked completely at every pair of adjacent times; finitely sampled events do not establish sigma-algebra inclusion.





## Symbol scope


- F — finite filtration
- t — time index





## Intuition, definition, and misconceptions

A finer partition represents more information.

Every later block must be contained in an earlier block.

An event is measurable exactly when it is a union of current blocks; adaptedness is checked separately.

An arbitrary partition sequence is not necessarily a filtration, and a filtration does not automatically make a process adapted, independent or Markov.





## Adopted derivation guide

[Finite filtration: checking information nesting by partition refinement, and separating it from adaptedness and independence](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-filtration-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/mko/mko-sigma-algebra-measure-space/) — Each time's information is a sigma-algebra on the same space.
- [Discrete-time stochastic process](https://ocme.evemisslab.com/en/mko/mko-discrete-stochastic-process/) — A filtration supplies time-indexed information semantics.
- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — The sigma-algebras are ordered in time.
- [Measurable function](https://ocme.evemisslab.com/en/mko/mko-measurable-function/) — Adaptedness separately requires each time's variable to be measurable.





## Verification boundary

Only complete finite partition sequences are handled; no natural-filtration or process-property claim is asserted.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Together with conditional expectation, add martingales.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/finite-filtration.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-filtration/index.md)