| import Mathlib.Probability.Independence.InfinitePi |
| import Mathlib.Probability.Kernel.Basic |
| import Mathlib.MeasureTheory.Measure.GiryMonad |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators |
|
|
| namespace Invariant233.KernelProduct |
|
|
| variable {ι : Type*} {A B : ι → Type*} |
| [∀ i, MeasurableSpace (A i)] [∀ i, MeasurableSpace (B i)] |
|
|
| def independent (κ : ∀ i, Kernel (A i) (B i)) [∀ i, IsMarkovKernel (κ i)] : |
| Kernel (∀ i, A i) (∀ i, B 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 _ => MeasurableSpace.isPiSystem_measurableSet) (by simp)) |
| rintro _ ⟨s, t, ht, rfl⟩ |
| simp only [Set.mem_univ_pi, Set.mem_ofPred_eq] at ht |
| simp_rw [Measure.infinitePi_pi _ (fun i _ => ht i)] |
| exact Finset.measurable_fun_prod s (fun i _ => |
| (Kernel.measurable_coe (κ i) (ht i)).comp (measurable_pi_apply i)) |
|
|
| end Invariant233.KernelProduct |
|
|
| namespace Invariant233.KernelProduct |
| variable {ι : Type*} {A B : ι → Type*} |
| [∀ i, MeasurableSpace (A i)] [∀ i, MeasurableSpace (B i)] |
|
|
| def pairCoordinates : ((∀ i, A i) × (∀ i, B i)) ≃ᵐ (∀ i, A i × B i) where |
| toFun p i := (p.1 i, p.2 i) |
| invFun z := (fun i => (z i).1, fun i => (z i).2) |
| left_inv p := rfl |
| right_inv z := rfl |
| measurable_toFun := by |
| change Measurable (fun p : (∀ i, A i) × (∀ i, B i) => fun i => (p.1 i, p.2 i)) |
| fun_prop |
| measurable_invFun := by |
| change Measurable (fun z : ∀ i, A i × B i => (fun i => (z i).1, fun i => (z i).2)) |
| fun_prop |
|
|
| end Invariant233.KernelProduct |
|
|
| end |
|
|
| end OAI |
|
|
