lean/OAI/Probability/FreeIsing/Conditioning/Descent.lean

import OAI.Probability.FreeIsing.Conditioning.Pullback

namespace OAI

noncomputable section

open MeasureTheory ProbabilityTheory

namespace Problem367

theorem condIndepFun_of_comp_measurePreserving
    {Ω Ω' C D : Type*} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω']
    [MeasurableSpace C] [MeasurableSpace D]
    [StandardBorelSpace Ω] [StandardBorelSpace Ω']
    (μ : Measure Ω) (ν : Measure Ω') [IsFiniteMeasure μ] [IsFiniteMeasure ν]
    (e : Ω → Ω') (he : MeasurePreserving e μ ν)
    (m : MeasurableSpace Ω') (hm : m ≤ mΩ')
    {f : Ω' → C} {g : Ω' → D}
    (hf : @Measurable Ω' C mΩ' _ f) (hg : @Measurable Ω' D mΩ' _ g)
    (hfg : CondIndepFun (m.comap e)
      ((MeasurableSpace.comap_mono hm).trans (@Measurable.comap_le Ω Ω' mΩ mΩ' e
        (@MeasurePreserving.measurable Ω Ω' mΩ mΩ' e μ ν he)))
      (f ∘ e) (g ∘ e) μ) : CondIndepFun m hm f g ν := by
  let : MeasurableSpace Ω' := mΩ'
  classical
  rw [condIndepFun_iff_condExp_inter_preimage_eq_mul hf hg]
  intro U V hU hV
  have hE := hU.preimage hf
  have hH := hV.preimage hg
  have hCE := condExp_comp_measurePreserving μ ν e he m hm
    ((integrable_const (1 : ℝ)).indicator hE)
  have hCH := condExp_comp_measurePreserving μ ν e he m hm
    ((integrable_const (1 : ℝ)).indicator hH)
  have hCEH := condExp_comp_measurePreserving μ ν e he m hm
    ((integrable_const (1 : ℝ)).indicator (hE.inter hH))
  have hmul := (condIndepFun_iff_condExp_inter_preimage_eq_mul
    (hf.comp he.measurable) (hg.comp he.measurable)).mp hfg U V hU hV
  have hmeas : MeasurableSet {ω' | (ν⟦f ⁻¹' U ∩ g ⁻¹' V | m⟧) ω' =
      (ν⟦f ⁻¹' U | m⟧) ω' * (ν⟦g ⁻¹' V | m⟧) ω'} :=
    measurableSet_eq_fun (stronglyMeasurable_condExp.mono hm).measurable
      ((stronglyMeasurable_condExp.mono hm).measurable.mul
        (stronglyMeasurable_condExp.mono hm).measurable)
  have hdesc := ae_map_iff (μ := μ) he.measurable.aemeasurable hmeas
  rw [he.map_eq] at hdesc
  apply hdesc.mpr
  filter_upwards [hCE, hCH, hCEH, hmul] with ω hωE hωH hωEH hωmul
  exact hωEH.symm.trans (hωmul.trans (congrArg₂ (· * ·) hωE hωH))

theorem condIndepFun_comp_measurePreserving_iff
    {Ω Ω' C D : Type*} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω']
    [MeasurableSpace C] [MeasurableSpace D]
    [StandardBorelSpace Ω] [StandardBorelSpace Ω']
    (μ : Measure Ω) (ν : Measure Ω') [IsFiniteMeasure μ] [IsFiniteMeasure ν]
    (e : Ω → Ω') (he : MeasurePreserving e μ ν)
    (m : MeasurableSpace Ω') (hm : m ≤ mΩ')
    {f : Ω' → C} {g : Ω' → D}
    (hf : @Measurable Ω' C mΩ' _ f) (hg : @Measurable Ω' D mΩ' _ g) :
    CondIndepFun (m.comap e)
      ((MeasurableSpace.comap_mono hm).trans (@Measurable.comap_le Ω Ω' mΩ mΩ' e
        (@MeasurePreserving.measurable Ω Ω' mΩ mΩ' e μ ν he)))
      (f ∘ e) (g ∘ e) μ ↔ CondIndepFun m hm f g ν := by
  let : MeasurableSpace Ω' := mΩ'
  exact ⟨condIndepFun_of_comp_measurePreserving μ ν e he m hm hf hg,
    condIndepFun_comp_measurePreserving μ ν e he m hm hf hg⟩

end Problem367

end

end OAI