`Learning.IT.filtrationAction`🔗

Definition

Filtration generated by the history at time n-1 together with the action at time n.

def

Learning.IT.filtrationAction.{u_4, u_5} (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] :
  MeasureTheory.Filtration ℕ inferInstance
Learning.IT.filtrationAction.{u_4, u_5}
  (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] :
  MeasureTheory.Filtration ℕ inferInstance

Code

def filtrationAction (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] :
    Filtration ℕ (inferInstance : MeasurableSpace (ℕ → 𝓐 × 𝓨)) where
  seq n := if n = 0 then MeasurableSpace.comap (action 0) inferInstance
    else IT.filtration 𝓐 𝓨 (n - 1) ⊔ MeasurableSpace.comap (action n) inferInstance
  mono' n m hnm := by
    simp only
    by_cases hn : n = 0
    · by_cases hm : m = 0
      · simp [hn, hm]
      · simp only [hn, ↓reduceIte, hm]
        refine le_sup_of_le_left ?_
        rw [← measurable_iff_comap_le]
        suffices Measurable[IT.filtration 𝓐 𝓨 0] (action 0) from
          this.mono ((IT.filtration 𝓐 𝓨).mono zero_le) le_rfl
        exact adapted_action 0
    have hm : m ≠ 0 := by grind
    simp only [hn, hm, ↓reduceIte]
    have hnm' : n - 1 ≤ m - 1 := by grind
    simp only [sup_le_iff]
    constructor
    · refine le_sup_of_le_left ?_
      exact (IT.filtration 𝓐 𝓨).mono hnm'
    · rcases eq_or_lt_of_le hnm with rfl | hlt
      · exact le_sup_of_le_right le_rfl
      refine le_sup_of_le_left ?_
      rw [← measurable_iff_comap_le]
      have h_le : n ≤ m - 1 := by grind
      suffices Measurable[IT.filtration 𝓐 𝓨 n] (action n) from
        this.mono ((IT.filtration 𝓐 𝓨).mono h_le) le_rfl
      exact adapted_action n
  le' n := by
    by_cases hn : n = 0
    · simp only [hn, ↓reduceIte]
      rw [← measurable_iff_comap_le]
      fun_prop
    simp only [hn, ↓reduceIte, sup_le_iff]
    constructor
    · exact (IT.filtration 𝓐 𝓨).le _
    · rw [← measurable_iff_comap_le]
      fun_prop

Actions: Source · Open Issue

Meaning unchanged since v4.33.0-rc1-29-gce231eb, the oldest revision on record (2026-07-30).

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 2 project declarations, 27 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Learning.IT.filtrationAction🔗

`Learning.IT.filtrationAction`🔗