completedIntegrand_stronglyProgressive

Plain-English statement

- Completion preserves strong progressiveness.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem completedIntegrand_stronglyProgressive
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : LocalProgressiveL2Integrand filtration mu T) :
    IsStronglyProgressive filtration (completedIntegrand hUsual eta) := by
  classical
  intro terminal
  have hbad : @MeasurableSet (Set.Iic terminal × Omega)
      (Subtype.instMeasurableSpace.prod (filtration terminal))
      {p | p.2 ∈ badEnergySet eta} :=
    (measurableSet_badEnergySet hUsual eta terminal).preimage measurable_snd
  have hite : @StronglyMeasurable (Set.Iic terminal × Omega) ℝ inferInstance
      (Subtype.instMeasurableSpace.prod (filtration terminal))
      (fun p => if p.2 ∈ badEnergySet eta then 0 else eta.process p.1 p.2) :=
    StronglyMeasurable.ite hbad stronglyMeasurable_const (eta.progressive terminal)
  simpa only [completedIntegrand] using hite

/-- Every completed sample path has an integrable square on the finite time
horizon. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedIntegrand.lean:36published source at 7bcd37294df1

Proof architecture

replace nonintegrable sample paths by zero while preserving the progressive sigma-algebra

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
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Why the statement has this shape

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Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Integrability is an input or proved output; a displayed integral alone does not supply it.