`ProbabilityTheory.iIndepFun_nat_iff_forall_indepFun`🔗

Lemma

No docstring.

theorem

ProbabilityTheory.iIndepFun_nat_iff_forall_indepFun.{u_2, u_4}
  {Ω : Type u_2} {E : Type u_4} {mΩ : MeasurableSpace Ω}
  {mE : MeasurableSpace E} {μ : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure μ] {X : ℕ → Ω → E}
  (hX : ∀ (n : ℕ), AEMeasurable (X n) μ) :
  iIndepFun X μ ↔
    ∀ (n : ℕ), IndepFun (X (n + 1)) (fun ω i => X (↑i) ω) μ
ProbabilityTheory.iIndepFun_nat_iff_forall_indepFun.{u_2,
    u_4}
  {Ω : Type u_2} {E : Type u_4}
  {mΩ : MeasurableSpace Ω}
  {mE : MeasurableSpace E}
  {μ : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure μ]
  {X : ℕ → Ω → E}
  (hX : ∀ (n : ℕ), AEMeasurable (X n) μ) :
  iIndepFun X μ ↔
    ∀ (n : ℕ),
      IndepFun (X (n + 1))
        (fun ω i => X (↑i) ω) μ

Code

lemma iIndepFun_nat_iff_forall_indepFun [IsProbabilityMeasure μ] {X : ℕ → Ω → E}
    (hX : ∀ n, AEMeasurable (X n) μ) :
    iIndepFun X μ ↔ ∀ n, X (n + 1) ⟂ᵢ[μ] fun ω (i : Iic n) ↦ X i ω

Proof

by
  constructor
  · intro h n
    exact (h.indepFun_finset₀ {n + 1} (Iic n) (by simp) hX).comp
      (measurable_pi_apply ⟨n + 1, by simp⟩) measurable_id
  · intro h
    rw [iIndepFun_iff_measure_inter_preimage_eq_mul]
    intro s sets hsets
    induction s using Finset.strongInductionOn with
    | _ s ih =>
    obtain rfl | hs := s.eq_empty_or_nonempty
    · simp
    · obtain hn_zero | hn_pos := (s.max' hs).eq_zero_or_pos
      · simp [eq_singleton_iff_unique_mem.mpr ⟨hn_zero ▸ max'_mem _ hs,
          fun j hj => Nat.le_zero.mp (hn_zero ▸ le_max' _ j hj)⟩]
      · have hs'_le : ∀ i ∈ s.erase (s.max' hs), i ∈ Iic (s.max' hs - 1) := fun i hi =>
          mem_Iic.mpr (Nat.lt_succ_iff.mp (Nat.succ_pred_eq_of_pos hn_pos ▸
            lt_max'_of_mem_erase_max' _ hs hi))
        let t : Set (Iic (s.max' hs - 1) → E) :=
          {f | ∀ i : s.erase (s.max' hs), f ⟨i.1, hs'_le i.1 i.2⟩ ∈ sets i.1}
        have ht : MeasurableSet t := by
          have : t = ⋂ i : s.erase (s.max' hs), (· ⟨i.1, hs'_le i.1 i.2⟩) ⁻¹' sets i.1 := by
            ext
            simp [t]
          exact this ▸ .iInter fun ⟨i, hi⟩ =>
            (hsets i (erase_subset _ _ hi)).preimage (measurable_pi_apply _)
        have heq : ⋂ i ∈ s.erase (s.max' hs), X i ⁻¹' sets i =
            (fun ω (j : Iic (s.max' hs - 1)) => X j ω) ⁻¹' t := by
          ext ω
          simp only [Set.mem_iInter, Set.mem_preimage, t]
          exact ⟨fun hω ⟨i, hi⟩ => hω i hi, fun hω i hi => hω ⟨i, hi⟩⟩
        have hind := h (s.max' hs - 1)
        rw [Nat.sub_add_cancel hn_pos] at hind
        rw [(insert_erase (max'_mem _ hs)).symm, set_biInter_insert, heq,
          hind.measure_inter_preimage_eq_mul _ _ (hsets _ (max'_mem _ hs)) ht, ← heq,
          ih _ (erase_ssubset (max'_mem _ hs)) fun i hi => hsets i (erase_subset _ _ hi),
          prod_insert (notMem_erase _ _)]

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 2th recorded change.

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

Dependency graph

Nothing to draw. Its statement rests on no other declaration in this project, and names nothing from a package left unaudited — so the graph is this declaration alone. That is the answer, not a missing picture.

Audit surface: 0 project declarations, 25 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.

ProbabilityTheory.iIndepFun_nat_iff_forall_indepFun🔗

`ProbabilityTheory.iIndepFun_nat_iff_forall_indepFun`🔗