| import Mathlib.Probability.ProductMeasure |
| import Mathlib.Probability.Independence.InfinitePi |
| import Mathlib.Probability.Independence.Integration |
| import Mathlib.Probability.Kernel.Composition.MeasureCompProd |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory ProbabilityTheory Finset MeasurableSpace |
| open scoped ENNReal |
| noncomputable section |
|
|
| / |
| finite cylinders and extended to the entire product sigma algebra. -/ |
| def independentKernel {ι : Type*} {X Y : ι → Type*} |
| [∀ i, MeasurableSpace (X i)] [∀ i, MeasurableSpace (Y i)] |
| (κ : ∀ i, Kernel (X i) (Y i)) [∀ i, IsMarkovKernel (κ i)] : |
| Kernel (∀ i, X i) (∀ i, Y i) where |
| toFun x := Measure.infinitePi (fun i => κ i (x i)) |
| measurable' := by |
| apply Measurable.measure_of_isPiSystem_of_isProbabilityMeasure |
| generateFrom_squareCylinders.symm |
| (isPiSystem_squareCylinders (fun _ => isPiSystem_measurableSet) (by simp)) |
| rintro _ ⟨I, S, hS, rfl⟩ |
| have hs (i : ι) : MeasurableSet (S i) := hS i (Set.mem_univ i) |
| simp_rw [Measure.infinitePi_pi _ (fun i _ => hs i)] |
| exact Finset.measurable_prod _ fun i _ => ((κ i).measurable_coe (hs i)).comp (measurable_pi_apply i) |
| |
| instance independentKernel_markov {ι : Type*} {X Y : ι → Type*} |
| [∀ i, MeasurableSpace (X i)] [∀ i, MeasurableSpace (Y i)] |
| (κ : ∀ i, Kernel (X i) (Y i)) [∀ i, IsMarkovKernel (κ i)] : |
| IsMarkovKernel (independentKernel κ) := ⟨fun x => by |
| change IsProbabilityMeasure (Measure.infinitePi (fun i => κ i (x i))) |
| infer_instance⟩ |
| |
| theorem independentKernel_pi {ι : Type*} {X Y : ι → Type*} |
| [∀ i, MeasurableSpace (X i)] [∀ i, MeasurableSpace (Y i)] |
| (κ : ∀ i, Kernel (X i) (Y i)) [∀ i, IsMarkovKernel (κ i)] |
| (x : ∀ i, X i) (I : Finset ι) (S : ∀ i, Set (Y i)) |
| (hS : ∀ i ∈ I, MeasurableSet (S i)) : |
| independentKernel κ x (Set.pi (I : Set ι) S) = ∏ i ∈ I, κ i (x i) (S i) := |
| Measure.infinitePi_pi _ hS |
| |
| theorem lintegral_infinitePi_finset_prod {ι : Type*} {X : ι → Type*} |
| [∀ i, MeasurableSpace (X i)] (μ : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (μ i)] |
| (I : Finset ι) (f : ∀ i, X i → ℝ≥0∞) (hf : ∀ i, Measurable (f i)) : |
| (∫⁻ x, ∏ i ∈ I, f i (x i) ∂Measure.infinitePi μ) = ∏ i ∈ I, ∫⁻ t, f i t ∂μ i := by |
| rw [lintegral_prod_eq_prod_lintegral_of_indepFun I |
| (fun (i : ι) (x : ∀ i, X i) => f i (x i)) (iIndepFun_infinitePi (P := μ) (X := f) hf) |
| (fun i => (hf i).comp (measurable_pi_apply i))] |
| apply Finset.prod_congr rfl |
| intro i _ |
| exact (measurePreserving_eval_infinitePi μ i).lintegral_comp (hf i) |
| |
| / |
| The equality is of measures on all measurable sets, not just rectangles. -/ |
| theorem independentKernel_compProd {ι : Type*} {X Y : ι → Type*} |
| [∀ i, MeasurableSpace (X i)] [∀ i, MeasurableSpace (Y i)] |
| (μ : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (μ i)] |
| (κ : ∀ i, Kernel (X i) (Y i)) [∀ i, IsMarkovKernel (κ i)] : |
| ((Measure.infinitePi μ) ⊗ₘ (independentKernel κ)).map |
| (fun z i => (z.1 i, z.2 i)) = Measure.infinitePi (fun i => (μ i) ⊗ₘ (κ i)) := by |
| have hm : Measurable (fun z : (∀ i, X i) × (∀ i, Y i) => fun i => (z.1 i, z.2 i)) := by |
| fun_prop |
| apply Measure.eq_infinitePi |
| intro I S hS |
| rw [Measure.map_apply hm (MeasurableSet.pi I.countable_toSet (fun i _ => hS i)), |
| Measure.compProd_apply (hm (MeasurableSet.pi I.countable_toSet (fun i _ => hS i)))] |
| have he (x : ∀ i, X i) : Prod.mk x ⁻¹' ((fun (z : (∀ i, X i) × (∀ i, Y i)) i => (z.1 i, z.2 i)) ⁻¹' Set.pi (I : Set ι) S) = |
| Set.pi (I : Set ι) (fun i => Prod.mk (x i) ⁻¹' S i) := rfl |
| simp_rw [he, independentKernel_pi κ _ I _ (fun i _ => measurable_prodMk_left (hS i))] |
| rw [lintegral_infinitePi_finset_prod μ I |
| (fun i t => κ i t (Prod.mk t ⁻¹' S i)) (fun i => Kernel.measurable_kernel_prodMk_left (hS i))] |
| apply Finset.prod_congr rfl |
| intro i _ |
| exact (Measure.compProd_apply (hS i)).symm |
| |
| end |
| end FiniteFactor |
| |
| end OAI |
| |
