File size: 2,882 Bytes
f1f7953 | 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 | import OAI.Analysis.IrrationalRotation.Couplings.ConditionalObservationTransport
import OAI.Analysis.IrrationalRotation.Sampling.MixedRadixHaarLaw
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped MeasureTheory
namespace Invariant233.MixedRadix
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω}
{r : ℕ → ℕ}
theorem ae_circleCanonical_circleValue_of_map_ac
(hr : ∀ n, 2 ≤ r n) (hp : ∀ n, 0 < r n)
{D : Ω → Digits r} (hD : Measurable D)
(hAC : μ.map D ≪ digitLaw r hp) :
∀ᵐ ω ∂μ, circleCanonical r hp (circleValue (D ω)) = D ω := by
exact ae_of_ae_map hD.aemeasurable
(hAC.ae_le (ae_circleCanonical_circleValue hr hp))
variable [IsFiniteMeasure μ]
theorem condExp_circleValue_eq_digits
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
(hr : ∀ n, 2 ≤ r n) (hp : ∀ n, 0 < r n)
{D : Ω → Digits r} (hD : Measurable D)
(hAC : μ.map D ≪ digitLaw r hp) (f : Ω → E) :
μ[f | MeasurableSpace.comap (circleValue ∘ D) inferInstance] =ᵐ[μ]
μ[f | MeasurableSpace.comap D inferInstance] :=
ConditionalObservationTransport.condExp_encoding hD (measurable_circleValue hr)
(measurable_circleCanonical r hp)
(ae_circleCanonical_circleValue_of_map_ac hr hp hD hAC) f
theorem condExp_circleValue_coordinate_formula
(hr : ∀ n, 2 ≤ r n) (hp : ∀ n, 0 < r n)
{D : Ω → Digits r} (hD : Measurable D)
(hAC : μ.map D ≪ digitLaw r hp) (i : ℕ)
(f : Ω → ℝ) (posterior : Fin (r i) → ℝ)
(hposterior : μ[f | MeasurableSpace.comap D inferInstance] =ᵐ[μ]
fun ω => posterior (D ω i)) :
μ[f | MeasurableSpace.comap (circleValue ∘ D) inferInstance] =ᵐ[μ]
fun ω => posterior (circleCanonical r hp (circleValue (D ω)) i) :=
ConditionalObservationTransport.condExp_encoding_formula hD (measurable_circleValue hr)
(measurable_circleCanonical r hp)
(ae_circleCanonical_circleValue_of_map_ac hr hp hD hAC) f
(fun d => posterior (d i)) hposterior
theorem iCondIndepFun_circleValue_iff_digits
[StandardBorelSpace Ω] {ι : Type*} {β : ι → Type*}
[∀ i, MeasurableSpace (β i)]
(hr : ∀ n, 2 ≤ r n) (hp : ∀ n, 0 < r n)
{D : Ω → Digits r} (hD : Measurable D)
(hAC : μ.map D ≪ digitLaw r hp)
(L : ∀ i, Ω → β i) (hL : ∀ i, Measurable (L i)) :
iCondIndepFun (MeasurableSpace.comap (circleValue ∘ D) inferInstance)
((measurable_circleValue hr).comp hD).comap_le L μ ↔
iCondIndepFun (MeasurableSpace.comap D inferInstance) hD.comap_le L μ :=
ConditionalObservationTransport.iCondIndepFun_encoding_iff hD (measurable_circleValue hr)
(measurable_circleCanonical r hp)
(ae_circleCanonical_circleValue_of_map_ac hr hp hD hAC) L hL
end Invariant233.MixedRadix
end
end OAI
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