cond_iInter — Mathlib

English

AEMeasurable version of the Div Div independence: independence holds for quotients under AEMeasurability assumptions.

Русский

Версия с AEMeasurable: независимость дробей сохраняется при соответствующих предпосылках.

LaTeX

$$$\\forall i,j,k,l,\\ Ne(i,k) \\land Ne(i,l) \\land Ne(j,k) \\land Ne(j,l)\\Rightarrow IndepFun(\\frac{f_i}{f_j}, \\frac{f_k}{f_l})\\mu$$$

Lean4

/-- The probability of an intersection of preimages conditioning on another intersection factors
into a product. -/
theorem cond_iInter [Finite ι] (hY : ∀ i, Measurable (Y i)) (hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) μ)
    (hf : ∀ i ∈ s, MeasurableSet[mα.comap (X i)] (f i)) (hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0)
    (ht : ∀ i, MeasurableSet (t i)) : μ[⋂ i ∈ s, f i | ⋂ i, Y i ⁻¹' t i] = ∏ i ∈ s, μ[f i | Y i in t i] :=
  by
  have : IsProbabilityMeasure (μ : Measure Ω) := hindep.isProbabilityMeasure
  classical
  cases nonempty_fintype ι
  let g (i' : ι) := if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
  calc
    _ = (μ (⋂ i, Y i ⁻¹' t i))⁻¹ * μ ((⋂ i, Y i ⁻¹' t i) ∩ ⋂ i ∈ s, f i) := by rw [cond_apply];
      exact .iInter fun i ↦ hY i (ht i)
    _ = (μ (⋂ i, Y i ⁻¹' t i))⁻¹ * μ (⋂ i, g i) := by
      congr
      calc
        _ = (⋂ i, Y i ⁻¹' t i) ∩ ⋂ i, if i ∈ s then f i else .univ := by
          simp only [Set.iInter_ite, Set.iInter_univ, Set.inter_univ]
        _ = ⋂ i, Y i ⁻¹' t i ∩ (if i ∈ s then f i else .univ) := by rw [Set.iInter_inter_distrib]
        _ = _ := Set.iInter_congr fun i ↦ by by_cases hi : i ∈ s <;> simp [hi, g]
    _ = (∏ i, μ (Y i ⁻¹' t i))⁻¹ * μ (⋂ i, g i) :=
      by
      rw [hindep.meas_iInter]
      exact fun i ↦ ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp⟩
    _ = (∏ i, μ (Y i ⁻¹' t i))⁻¹ * ∏ i, μ (g i) :=
      by
      rw [hindep.meas_iInter]
      intro i
      by_cases hi : i ∈ s <;> simp only [hi, ↓reduceIte, g]
      · obtain ⟨A, hA, hA'⟩ := hf i hi
        exact
          .inter ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp⟩
            ⟨A ×ˢ Set.univ, hA.prod .univ, by ext; simp [← hA']⟩
      · exact ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp⟩
    _ = ∏ i, (μ (Y i ⁻¹' t i))⁻¹ * μ (g i) :=
      by
      rw [Finset.prod_mul_distrib, ENNReal.prod_inv_distrib]
      exact fun _ _ i _ _ ↦ .inr <| measure_ne_top _ _
    _ = ∏ i, if i ∈ s then μ[f i | Y i ⁻¹' t i] else 1 :=
      by
      refine Finset.prod_congr rfl fun i _ ↦ ?_
      by_cases hi : i ∈ s
      · simp only [hi, ↓reduceIte, g, cond_apply (hY i (ht i))]
      · simp only [hi, ↓reduceIte, g, ENNReal.inv_mul_cancel (hy i hi) (measure_ne_top μ _)]
    _ = _ := by simp

Похожие утверждения