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[Merged by Bors] - feat: the natural filtration of a process is the filtration generated by the process #39754

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[Merged by Bors] - feat: the natural filtration of a process is the filtration generated by the process #39754

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Automated addition of changes in Mathlib/MeasureTheory/MeasurableSpac…
EtienneC30 May 22, 2026
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feat: the natural filtration is the filtration generated by the process
EtienneC30 May 23, 2026
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6 changes: 6 additions & 0 deletions Mathlib/MeasureTheory/MeasurableSpace/Constructions.lean
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Expand Up @@ -586,6 +586,12 @@ theorem measurable_pi_lambda (f : α → ∀ a, X a) (hf : ∀ a, Measurable fun
Measurable f :=
measurable_pi_iff.mpr hf

lemma MeasurableSpace.comap_process_pi (X : (a : δ) → β → X a) :
MeasurableSpace.comap (fun b a ↦ X a b) inferInstance =
⨆ a, MeasurableSpace.comap (X a) inferInstance := by
simp_rw [MeasurableSpace.pi, MeasurableSpace.comap_iSup, MeasurableSpace.comap_comp]
rfl

/-- The function `(f, x) ↦ update f a x : (Π a, X a) × X a → Π a, X a` is measurable. -/
@[fun_prop]
theorem measurable_update' {a : δ} [DecidableEq δ] :
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5 changes: 5 additions & 0 deletions Mathlib/Probability/Process/Filtration.lean
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Expand Up @@ -399,6 +399,11 @@ def natural (u : (i : ι) → Ω → β i) (hum : ∀ i, StronglyMeasurable (u i
rintro j _ s ⟨t, ht, rfl⟩
exact (hum j).measurable ht

lemma natural_eq_comap (u : (i : ι) → Ω → β i) (hum : ∀ (i : ι), StronglyMeasurable (u i)) (i : ι) :
natural u hum i = .comap (fun ω (j : Set.Iic i) ↦ u j ω) inferInstance := by
simp_rw [natural, MeasurableSpace.comap_process_pi, iSup_subtype']
rfl

section

open MeasurableSpace
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