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9eeb9c7 | 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 | 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
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