Plain-English statement
- A real Brownian motion relative to a specified filtration. The last field is the condition needed for stochastic integration: the increment after `s` is independent of the whole past sigma-algebra `F_s`. Bare independent increments do not imply this for an arbitrary enlarged filtration.
Read the mathematics first, then descend into Lean
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Proof architecture
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How to read the exact Lean declaration
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- NameWhat reusable mathematical fact is being created?
- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
- PropositionAfter the colon, translate the Lean expression back into a paper statement.
- Proof actionsAfter
by, ask what each tactic does to the mathematical goal—not only what syntax it uses.
Syntax used on this page
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Lean statement
structure IsBrownianMotionWithFiltration
{Omega : Type*} {m : MeasurableSpace Omega}
(B : ℝ≥0 → Omega → ℝ) (filtration : Filtration ℝ≥0 m)
(mu : Measure Omega) : Prop where
isBrownian : ProbabilityTheory.IsBrownianReal B mu
stronglyAdapted : StronglyAdapted filtration B
incrementIndependent : ∀ s t, s ≤ t →
Indep (filtration s)
(MeasurableSpace.comap (fun omega => B t omega - B s omega) (borel ℝ)) mu
namespace IsBrownianMotionWithFiltration
variable {Ω : Type*} {m : MeasurableSpace Ω}
{B : ℝ≥0 → Ω → ℝ} {filtration : Filtration ℝ≥0 m} {μ : Measure Ω}
/-- A Brownian-filtration contract carries a probability measure. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:64published source at 7bcd37294df1
Proof architecture
ensure future Brownian increments are independent of the whole past sigma-algebra used by adapted coefficients
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- The declaration is definition-like or term-style; its typed right-hand side is the proof object.
Why the statement has this shape
The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.
Hidden assumptions and non-claims
- Measurability is represented explicitly or must be supplied by a dependency.