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import OAI.Analysis.CoulombIonization.Localization.UnorderedObservations
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff
open MeasureTheory Filter
namespace CoulombObservation
lemma condExp_observable_sandwich {Ω : Type*} {m₀ : MeasurableSpace Ω}
{m : MeasurableSpace Ω} (hm : m ≤ m₀) (μ : @Measure Ω m₀) [IsFiniteMeasure μ]
{f h d : Ω → ℝ} (hf : Integrable f μ) (hh : Integrable h μ) (hd : Integrable d μ)
(hhm : StronglyMeasurable[m] h) (hdm : StronglyMeasurable[m] d)
(hb : ∀ᵐ z ∂μ, |f z-h z| ≤ d z) :
∀ᵐ z ∂μ, |μ[f | m] z-h z| ≤ d z := by
have hlow : (fun z => h z-d z) ≤ᵐ[μ] f := hb.mono (fun z hz => by
have := (abs_le.mp hz).1
linarith)
have hupp : f ≤ᵐ[μ] (fun z => h z+d z) := hb.mono (fun z hz => by
have := (abs_le.mp hz).2
linarith)
have hlo := condExp_mono (m := m) (hh.sub hd) hf hlow
have hup := condExp_mono (m := m) hf (hh.add hd) hupp
rw [condExp_of_stronglyMeasurable (m₀ := m₀) (μ := μ) hm (hhm.sub hdm) (hh.sub hd)] at hlo
rw [condExp_of_stronglyMeasurable (m₀ := m₀) (μ := μ) hm (hhm.add hdm) (hh.add hd)] at hup
filter_upwards [hlo,hup] with z hl hu
simp only [Pi.sub_apply,Pi.add_apply] at hl hu
exact abs_le.mpr ⟨by linarith,by linarith⟩
lemma raw_condExp_observable_error {Ω : Type*} {m₀ : MeasurableSpace Ω}
{m : MeasurableSpace Ω} (hm : m ≤ m₀) (μ : @Measure Ω m₀) [IsFiniteMeasure μ]
{f h d : Ω → ℝ} (hf : Integrable f μ) (hh : Integrable h μ) (hd : Integrable d μ)
(hhm : StronglyMeasurable[m] h) (hdm : StronglyMeasurable[m] d)
(hb : ∀ᵐ z ∂μ, |f z-h z| ≤ d z) :
∀ᵐ z ∂μ, |f z-μ[f | m] z| ≤ 2*d z := by
filter_upwards [hb,condExp_observable_sandwich hm μ hf hh hd hhm hdm hb] with z hz hc
have he : f z-μ[f | m] z = (f z-h z)+(h z-μ[f | m] z) := by ring
rw [he]
calc
_ ≤ |f z-h z|+|h z-μ[f | m] z| := abs_add_le _ _
_ ≤ d z+d z := add_le_add hz (by rwa [abs_sub_comm])
_ = _ := by ring
end CoulombObservation
open MeasureTheory Filter Set Metric
open scoped BigOperators NNReal
namespace CoulombAtom
open CoulombObservation
lemma integrable_configuration_test {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ] {N : ℕ} {X : Ω → Configuration N}
(hX : Measurable X) {g : Space → ℝ} (hg : Continuous g) (hc : HasCompactSupport g) :
Integrable (fun z => ∑ i, g (X z i)) μ := by
obtain ⟨v,hv⟩ := hg.norm.exists_forall_ge_of_hasCompactSupport hc.norm
apply integrable_finsetSum
intro i _
apply (integrable_const ‖g v‖).mono'
((hg.measurable.comp ((measurable_pi_apply i).comp hX)).aestronglyMeasurable)
exact ae_of_all _ (fun z => hv (X z i))
lemma observedLocalCount_integrable {N K : ℕ} (μ : Measure (Configuration N))
[IsFiniteMeasure μ] (ell : Fin K → ℝ) (k : Fin K) (y : Space) (R : ℝ) :
Integrable (observedLocalCount (N := N) ell k y R) (physicalObservationLaw μ K) := by
have h := (observedLocalCount_information_measurable (N := N) ell k (Nat.zero_le _) y R).mono
(observationInformation_le ell 0) (le_refl _)
apply (integrable_const (N:ℝ)).mono' h.aestronglyMeasurable
exact ae_of_all _ (fun z => by
simpa only [Real.norm_eq_abs,abs_of_nonneg (observedLocalCount_nonneg ell k y R z)] using
observedLocalCount_le ell k y R z)
def posteriorTestValue {N K : ℕ} (μ : Measure (Configuration N)) (ell : Fin K → ℝ)
(j : ℕ) (g : Space → ℝ) :
(Configuration N × (Fin K × (Fin N × Fin 3) → ℝ)) → ℝ :=
(physicalObservationLaw μ K)[(fun z => ∑ i, g (z.1 i)) | observationInformation ell j]
lemma posteriorTestValue_tower {N K : ℕ} (μ : Measure (Configuration N))
[IsFiniteMeasure μ] (ell : Fin K → ℝ) {j k : ℕ} (hjk : j ≤ k) (g : Space → ℝ) :
(physicalObservationLaw μ K)[posteriorTestValue μ ell j g | observationInformation ell k] =ᵐ[
physicalObservationLaw μ K] posteriorTestValue μ ell k g :=
condExp_condExp_of_le (observationInformation_antitone ell hjk) (observationInformation_le ell j)
theorem posterior_observed_test_sandwich {N K : ℕ} (μ : Measure (Configuration N))
[IsFiniteMeasure μ] (ell : Fin K → ℝ) (hell : ∀ k, 0 ≤ ell k)
{j : ℕ} (k : Fin K) (hk : j ≤ k.val) {g : Space → ℝ} {L : ℝ≥0}
(hg : LipschitzWith L g) (y : Space) (R : ℝ)
(hs : Function.support g ⊆ closedBall y R) :
∀ᵐ z ∂physicalObservationLaw μ K,
|posteriorTestValue μ ell j g z-(∑ i, g (physicalObservedConfiguration ell k z i))| ≤
observedLocalCount ell k y (R+Real.sqrt 3*ell k) z*((L:ℝ)*(Real.sqrt 3*ell k)) := by
have hc : HasCompactSupport g := HasCompactSupport.of_support_subset_isCompact (isCompact_closedBall y R) hs
have hf := integrable_configuration_test (physicalObservationLaw μ K)
measurable_fst hg.continuous hc
have hh := integrable_configuration_test (physicalObservationLaw μ K)
(physicalObservedConfiguration_measurable ell k) hg.continuous hc
have hd := (observedLocalCount_integrable μ ell k y (R+Real.sqrt 3*ell k)).mul_const
((L:ℝ)*(Real.sqrt 3*ell k))
have hhm := (observedStatistic_information_measurable (N := N) ell k hk hg.continuous.measurable).stronglyMeasurable
have hdm := ((observedLocalCount_information_measurable (N := N) ell k hk y (R+Real.sqrt 3*ell k)).mul_const
((L:ℝ)*(Real.sqrt 3*ell k))).stronglyMeasurable
apply condExp_observable_sandwich (observationInformation_le ell j)
(physicalObservationLaw μ K) hf hh hd hhm hdm
filter_upwards [physicalObservationLaw_ae_displacement μ ell hell] with z hz
exact matchedLocal_test_error z.1 (physicalObservedConfiguration ell k z)
(mul_nonneg (Real.sqrt_nonneg _) (hell k)) (hz k) hg y hs
theorem raw_posterior_local_test_error {N K : ℕ} (μ : Measure (Configuration N))
[IsFiniteMeasure μ] (ell : Fin K → ℝ) (hell : ∀ k, 0 ≤ ell k)
{j : ℕ} (k : Fin K) (hk : j ≤ k.val) {g : Space → ℝ} {L : ℝ≥0}
(hg : LipschitzWith L g) (y : Space) (R : ℝ)
(hs : Function.support g ⊆ closedBall y R) :
∀ᵐ z ∂physicalObservationLaw μ K,
|(∑ i, g (z.1 i))-posteriorTestValue μ ell j g z| ≤
2*observedLocalCount ell k y (R+Real.sqrt 3*ell k) z*((L:ℝ)*(Real.sqrt 3*ell k)) := by
filter_upwards [physicalObservationLaw_ae_displacement μ ell hell,
posterior_observed_test_sandwich μ ell hell k hk hg y R hs] with z hz hp
have hr := matchedLocal_test_error z.1 (physicalObservedConfiguration ell k z)
(mul_nonneg (Real.sqrt_nonneg _) (hell k)) (hz k) hg y hs
have he : (∑ i, g (z.1 i))-posteriorTestValue μ ell j g z =
((∑ i, g (z.1 i))-(∑ i, g (physicalObservedConfiguration ell k z i)))+
((∑ i, g (physicalObservedConfiguration ell k z i))-posteriorTestValue μ ell j g z) := by ring
rw [he]
refine (abs_add_le _ _).trans ?_
rw [abs_sub_comm (∑ i, g (physicalObservedConfiguration ell k z i))]
change _+_ ≤ 2*observedLocalCount ell k y (R+Real.sqrt 3*ell k) z*((L:ℝ)*(Real.sqrt 3*ell k))
dsimp only [observedLocalCount] at hp ⊢
linarith
end CoulombAtom
end
end OAI
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