DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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.

The Brownian germ sigma-algebra at zero

Definition

Assume the Axiom of Choice and let $B$ be a standard Brownian motion with raw natural filtration $(F_{t}^{0})$ and usual augmentation $(F_{t})$ Natural and usual augmented Brownian filtrations. The germ sigma-algebra at zero is $F_{0 +}^{0} : = ⋂_{t > 0} F_{t}^{0} .$ It records the events observable at arbitrarily small positive times in the uncompleted filtration.

Three elementary descriptions are part of the definition.

Countable intersection. Since $t \mapsto F_{t}^{0}$ is increasing, $F_{0 +}^{0} = ⋂_{q \in Q, q > 0} F_{q}^{0}$ . Indeed every positive rational is a positive real, giving one inclusion, while for a real $t > 0$ one may choose a rational $q \in (0, t)$ and use $F_{q}^{0} \subseteq F_{t}^{0}$ , so the countable intersection is contained in every $F_{t}^{0}$ .
Position relative to the usual augmentation. $F_{0 +}^{0} \subseteq F_{0}$ . For $u > 0$ one has $F_{u / 2}^{0} \subseteq F_{u}^{0} \subseteq \overline{F}_{u}^{0}$ , so $F_{0 +}^{0} \subseteq F_{u / 2}^{0} \subseteq \overline{F}_{u}^{0}$ for every $u > 0$ ; intersecting over $u > 0$ gives the claim. In particular every germ event is an event of the usual sigma-algebra at time zero.
No completion is included. The definition uses the raw sigma-algebras. The completed and augmented objects of Natural and usual augmented Brownian filtrations are not substituted for them, and the companion examples page records a counterexample showing that $F_{0 +}^{0} \neq F_{0}^{0}$ in the canonical realization.

AC is declared only because the ambient filtration definition inherits the Brownian construction's assumption; the intersection and its two descriptions make no selection.

Source notes

Sousi, Definition 6.10 and Theorem 6.13, and Durrett, Theorem 7.2.3, use the germ sigma-algebra at zero as the home of Blumenthal's zero-one law. The countable-intersection description is the form in which the increasing filtration is used, and the containment in the time-zero usual sigma-algebra is recorded because the zero-one law consumes it.

Depends on

Used by

The raw natural Brownian filtration need not be right-continuous Counterexample
Blumenthal's zero-one law Theorem
Brownian-filtration martingale representation Theorem

Dependency tree · two levels

10 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

Perla Sousi, Advanced Probability, Definition 6.10 and Theorem 6.13 (standard reference, not scraped)
Rick Durrett, Probability: Theory and Examples, fifth edition, Theorem 7.2.3 (standard reference, not scraped)