| import Mathlib |
| import OAI.Analysis.CoulombIonization.FieldAnalysis.BarrierObservationTowerBarrier |
| import OAI.Analysis.CoulombIonization.RadialBounds.Error |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter Set Metric ProbabilityTheory |
| open scoped BigOperators Topology |
| namespace CoulombBarrier |
| open CoulombAtom CoulombAnalysis |
| variable {Ω D : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] |
| variable [MeasurableSpace D] (P : Measure Ω) [IsProbabilityMeasure P] |
| variable (X : Ω → D) (hX : Measurable X) |
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| lemma conditionalField_nonneg {p : Ω → TFSpace → ℝ} (hn : ∀ sample x, 0 ≤ p sample x) |
| (d : D) (x : TFSpace) : 0 ≤ conditionalField P X p d x := |
| integral_nonneg (fun sample => hn sample x) |
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| lemma conditionalField_support {p : Ω → TFSpace → ℝ} {R : ℝ} |
| (hs : ∀ sample x, R < ‖x‖ → p sample x = 0) (d : D) (x : TFSpace) (hx : R < ‖x‖) : |
| conditionalField P X p d x = 0 := by |
| simp only [conditionalField,hs _ x hx,integral_zero] |
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| include hX |
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| lemma conditionalField_expectation {p : Ω → TFSpace → ℝ} |
| (hm : Measurable (Function.uncurry p)) (hb : DeterministicLocalBound p) (x : TFSpace) : |
| (∫ sample, conditionalField P X p (X sample) x ∂P) = ∫ sample, p sample x ∂P := by |
| rw [←integral_congr_ae (conditionalField_eq_condExp P hX hm hb x)] |
| exact integral_condExp hX.comap_le |
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| theorem conditionalField_totalMass {p : Ω → TFSpace → ℝ} |
| (hm : Measurable (Function.uncurry p)) (hb : DeterministicLocalBound p) |
| {R : ℝ} (hs : ∀ sample x, R < ‖x‖ → p sample x = 0) : |
| (∫ sample, (∫ x, conditionalField P X p (X sample) x) ∂P) = ∫ sample, (∫ x, p sample x) ∂P := by |
| have hcm : Measurable (fun z : Ω × TFSpace => conditionalField P X p (X z.1) z.2) := |
| (conditionalField_measurable P X hm).comp ((hX.comp measurable_fst).prodMk measurable_snd) |
| have hcb : DeterministicLocalBound (fun sample x => conditionalField P X p (X sample) x) := by |
| intro K hK |
| obtain ⟨C,hC⟩ := conditionalField_bound P X hb K hK |
| exact ⟨C,fun sample x hx => hC (X sample) x hx⟩ |
| have hcs : ∀ sample x, R < ‖x‖ → conditionalField P X p (X sample) x = 0 := |
| fun sample x hx => conditionalField_support P X hs (X sample) x hx |
| rw [integral_integral_swap (compactBound_integrable_prod hcm hcb hcs), integral_integral_swap (compactBound_integrable_prod hm hb hs)] |
| exact integral_congr_ae (Eventually.of_forall fun x => conditionalField_expectation P X hX hm hb x) |
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| theorem conditionalField_spatial_bounds {u : Ω → TFSpace → ℝ} |
| (hm : Measurable (Function.uncurry u)) (hb : DeterministicLocalBound u) |
| (V a b : TFSpace → ℝ) (A : Set TFSpace) |
| (hlo : ∀ᵐ sample ∂P, ∀ x ∈ A, a x ≤ V x+u sample x) |
| (hhi : ∀ᵐ sample ∂P, ∀ x ∈ A, V x+u sample x ≤ b x) : |
| ∀ᵐ d ∂P.map X, ∀ x ∈ A, |
| a x ≤ V x+conditionalField P X u d x ∧ |
| V x+conditionalField P X u d x ≤ b x := by |
| filter_upwards [conditional_all_properties P hX hlo,conditional_all_properties P hX hhi] with d hdlo hdhi |
| intro x hx |
| have hi := (integrable_const (V x) : Integrable (fun _ : Ω => V x) (condDistrib id X P d)).add |
| (deterministicBound_integrable_section hm hb x) |
| have he : (∫ sample, (V x+u sample x) ∂condDistrib id X P d) = V x+conditionalField P X u d x := by |
| rw [integral_add (integrable_const _) (deterministicBound_integrable_section hm hb x)] |
| simp [conditionalField] |
| constructor |
| · rw [←he] |
| simpa only [integral_const,probReal_univ,one_smul,Pi.add_apply] using |
| integral_mono_ae (integrable_const (a x)) hi (hdlo.mono fun sample hω => hω x hx) |
| · rw [←he] |
| simpa only [integral_const,probReal_univ,one_smul,Pi.add_apply] using |
| integral_mono_ae hi (integrable_const (b x)) (hdhi.mono fun sample hω => hω x hx) |
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| end CoulombBarrier |
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| end |
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| end OAI |
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