DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Filtration and filtered probability space

Definition

A discrete filtered probability space is $(Ω, F, (F_{n})_{n \in N_{0}}, P)$ , where $(Ω, F, P)$ is a probability space Probability measures and probability spaces, each $F_{n}$ is a sigma-algebra on $Ω$ Sigma-algebras, and $F_{n} \subseteq F_{n + 1} \subseteq F$ . The sequence $(F_{n})$ is a filtration. Time starts at zero. Neither completeness nor triviality of $F_{0}$ is required.

Restriction of $P$ to $F_{n}$ has total mass one and is countably additive: a disjoint sequence in $F_{n}$ is also such a sequence in $F$ , with the same union and the same values of $P$ . Thus each restriction is a probability measure. This specifies a structure and uses no choice principle. Real-valued processes will be the default.

Depends on

Used by

Adapted and integrable stochastic process Definition
Predictable discrete time process Definition
Martingale differences are orthogonal in l2 Lemma

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

van der Vaart, Martingales, Diffusions and Financial Mathematics (standard reference, not scraped)