File size: 3,153 Bytes
f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 | 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
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