openAI-math-dataset / lean /OAI /Probability /FreeIsing /Conditioning /FamilyConditionalPullback.lean
| import OAI.Probability.FreeIsing.Conditioning.Pullback | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace Problem367 | |
| theorem iCondIndepFun_comp_measurePreserving | |
| {Ω Ω' ι : Type*} {C : ι → Type*} | |
| [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] | |
| [mC : ∀ i, MeasurableSpace (C i)] | |
| [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 : ∀ i, Ω' → C i} (hf : ∀ i, @Measurable Ω' (C i) mΩ' (mC i) (f i)) | |
| (hfamily : iCondIndepFun m hm f ν) : | |
| iCondIndepFun (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))) | |
| (fun i => f i ∘ e) μ := by | |
| let : MeasurableSpace Ω' := mΩ' | |
| classical | |
| rw [iCondIndepFun_iff_condExp_inter_preimage_eq_mul mC (fun i => f i ∘ e) | |
| (fun i => (hf i).comp he.measurable)] | |
| intro S sets hsets | |
| let E : Set Ω' := ⋂ i ∈ S, f i ⁻¹' sets i | |
| have hE : MeasurableSet E := | |
| S.measurableSet_biInter (fun i hi => (hsets i hi).preimage (hf i)) | |
| have hCE := condExp_comp_measurePreserving μ ν e he m hm | |
| ((integrable_const (1 : ℝ)).indicator hE) | |
| have heach : ∀ᵐ ω ∂μ, ∀ i ∈ S, | |
| μ[((f i ⁻¹' sets i).indicator (fun _ => (1 : ℝ))) ∘ e | m.comap e] ω = | |
| ν[(f i ⁻¹' sets i).indicator (fun _ => (1 : ℝ)) | m] (e ω) := by | |
| apply S.eventually_all.mpr | |
| intro i hi | |
| exact condExp_comp_measurePreserving μ ν e he m hm | |
| ((integrable_const (1 : ℝ)).indicator ((hsets i hi).preimage (hf i))) | |
| have hmul := he.quasiMeasurePreserving.ae_eq_comp | |
| ((iCondIndepFun_iff_condExp_inter_preimage_eq_mul mC f hf).mp hfamily S hsets) | |
| filter_upwards [hCE, heach, hmul] with ω hωE hωeach hωmul | |
| simp only [Finset.prod_apply, Function.comp_apply] at hωmul ⊢ | |
| have hindicator : (⋂ i ∈ S, (f i ∘ e) ⁻¹' sets i).indicator | |
| (fun _ => (1 : ℝ)) = (E.indicator (fun _ => (1 : ℝ))) ∘ e := by | |
| funext x | |
| simp [E, Set.indicator, Function.comp_def] | |
| rw [hindicator] | |
| exact hωE.trans (hωmul.trans (Finset.prod_congr rfl (fun i hi => (hωeach i hi).symm))) | |
| theorem iCondIndepFun_iff_comp_measurePreserving | |
| {Ω Ω' ι : Type*} {C : ι → Type*} | |
| [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] | |
| [mC : ∀ i, MeasurableSpace (C i)] | |
| [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 : ∀ i, Ω' → C i} (hf : ∀ i, @Measurable Ω' (C i) mΩ' (mC i) (f i)) : | |
| iCondIndepFun m hm f ν ↔ | |
| iCondIndepFun (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))) | |
| (fun i => f i ∘ e) μ := by | |
| let : MeasurableSpace Ω' := mΩ' | |
| classical | |
| refine ⟨iCondIndepFun_comp_measurePreserving μ ν e he m hm hf, ?_⟩ | |
| intro hsource | |
| rw [iCondIndepFun_iff_condExp_inter_preimage_eq_mul mC f hf] | |
| intro S sets hsets | |
| have hmCE (h : Ω' → ℝ) : Measurable (ν[h | m]) := | |
| (stronglyMeasurable_condExp.mono hm).measurable | |
| apply (ae_eq_iff_comp_measurePreserving μ ν e he (hmCE _) ?_).mpr | |
| swap | |
| · convert S.measurable_prod (fun i hi => | |
| hmCE ((f i ⁻¹' sets i).indicator (fun _ => (1 : ℝ)))) using 1 | |
| ext ω' | |
| simp only [Finset.prod_apply] | |
| let E : Set Ω' := ⋂ i ∈ S, f i ⁻¹' sets i | |
| have hE : MeasurableSet E := | |
| S.measurableSet_biInter (fun i hi => (hsets i hi).preimage (hf i)) | |
| have hCE := condExp_comp_measurePreserving μ ν e he m hm | |
| ((integrable_const (1 : ℝ)).indicator hE) | |
| have heach : ∀ᵐ ω ∂μ, ∀ i ∈ S, | |
| μ[((f i ⁻¹' sets i).indicator (fun _ => (1 : ℝ))) ∘ e | m.comap e] ω = | |
| ν[(f i ⁻¹' sets i).indicator (fun _ => (1 : ℝ)) | m] (e ω) := by | |
| apply S.eventually_all.mpr | |
| intro i hi | |
| exact condExp_comp_measurePreserving μ ν e he m hm | |
| ((integrable_const (1 : ℝ)).indicator ((hsets i hi).preimage (hf i))) | |
| have hmul := (iCondIndepFun_iff_condExp_inter_preimage_eq_mul mC | |
| (fun i => f i ∘ e) (fun i => (hf i).comp he.measurable)).mp hsource S hsets | |
| filter_upwards [hCE, heach, hmul] with ω hωE hωeach hωmul | |
| have hindicator : (⋂ i ∈ S, (f i ∘ e) ⁻¹' sets i).indicator | |
| (fun _ => (1 : ℝ)) = (E.indicator (fun _ => (1 : ℝ))) ∘ e := by | |
| funext x | |
| simp [E, Set.indicator, Function.comp_def] | |
| rw [hindicator] at hωmul | |
| simp only [Function.comp_apply, Finset.prod_apply] at hωmul ⊢ | |
| exact hωE.symm.trans (hωmul.trans (Finset.prod_congr rfl (fun i hi => hωeach i hi))) | |
| alias ⟨_, iCondIndepFun_of_comp_measurePreserving⟩ := | |
| iCondIndepFun_iff_comp_measurePreserving | |
| end Problem367 | |
| end | |
| end OAI | |