| import OAI.Probability.ClassicalON.UpperEndpoint |
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| namespace OAI |
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| universe uV |
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| noncomputable section |
| open MeasureTheory Set |
| open scoped BigOperators Topology Classical |
| namespace ClassicalON |
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| variable {V : Type uV} [Fintype V] |
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| def glueAmplitude (B : Set V) (a : B → Amplitude) (r : ↥(Bᶜ) → Amplitude) : V → Amplitude := |
| (MeasurableEquiv.piEquivPiSubtypeProd (fun _ : V => Amplitude) (·∈B)).symm (a,r) |
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| omit [Fintype V] in |
| theorem continuous_glueAmplitude (B : Set V) : |
| Continuous (fun p : (B → Amplitude)×(↥(Bᶜ) → Amplitude) => glueAmplitude B p.1 p.2) := |
| (Homeomorph.piEquivPiSubtypeProd (·∈B) (fun _ : V => Amplitude)).symm.continuous |
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| omit [Fintype V] in |
| @[simp] theorem glueAmplitude_in (B : Set V) (a : B → Amplitude) (r : ↥(Bᶜ) → Amplitude) |
| (v : B) : glueAmplitude B a r v=a v := by |
| simp [glueAmplitude,MeasurableEquiv.piEquivPiSubtypeProd,Equiv.piEquivPiSubtypeProd,v.property] |
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| omit [Fintype V] in |
| @[simp] theorem glueAmplitude_out (B : Set V) (a : B → Amplitude) (r : ↥(Bᶜ) → Amplitude) |
| (v : ↥(Bᶜ)) : glueAmplitude B a r v=r v := by |
| simp [glueAmplitude,MeasurableEquiv.piEquivPiSubtypeProd,Equiv.piEquivPiSubtypeProd,show v.val∉B from v.property] |
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| theorem integral_amplitude_split (μ : Measure Amplitude) [IsProbabilityMeasure μ] |
| (B : Set V) (F : (V → Amplitude) → ℝ) (hF : Continuous F) : |
| (∫ r,F r ∂Measure.pi (fun _ : V => μ))= |
| ∫ a,∫ r,F (glueAmplitude B a r) ∂Measure.pi (fun _ : ↥(Bᶜ) => μ) |
| ∂Measure.pi (fun _ : B => μ) := by |
| let e := MeasurableEquiv.piEquivPiSubtypeProd (fun _ : V => Amplitude) (·∈B) |
| have hp := (measurePreserving_piEquivPiSubtypeProd (fun _ : V => μ) (·∈B)).symm e |
| rw [← hp.map_eq,integral_map hp.measurable.aemeasurable hF.aestronglyMeasurable] |
| exact integral_prod _ (compact_integrable (hF.comp (continuous_glueAmplitude B))) |
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| theorem pinEndpoint_nonneg (μ : Measure Amplitude) [IsProbabilityMeasure μ] |
| [μ.IsOpenPosMeasure] (B : Set V) {F : (V → Amplitude) → ℝ} (hF : Continuous F) |
| (ht : ∀ h : (B → Amplitude) → ℝ,Continuous h → (∀ a,0≤h a) → Monotone h → |
| 0≤∫ r,h (fun v : B => r v)*F r ∂Measure.pi (fun _ : V => μ)) : |
| 0≤∫ r,F (glueAmplitude B ⊤ r) ∂Measure.pi (fun _ : ↥(Bᶜ) => μ) := by |
| have hc : Continuous (fun a => ∫ r,F (glueAmplitude B a r) ∂Measure.pi (fun _ : ↥(Bᶜ) => μ)) := |
| compact_parametric_integral _ (hF.comp (continuous_glueAmplitude B)) |
| apply upperEndpoint_nonneg (Measure.pi (fun _ : B => μ)) hc |
| intro h hh hn hm |
| have hh' : Continuous (fun r : V → Amplitude => h (fun v : B => r v)) := by |
| apply hh.comp |
| exact continuous_pi (fun v => continuous_apply v.val) |
| have hs := integral_amplitude_split μ B (fun r => h (fun v : B => r v)*F r) (hh'.mul hF) |
| have he : (fun a => ∫ r,h (fun v : B => glueAmplitude B a r v)*F (glueAmplitude B a r) |
| ∂Measure.pi (fun _ : ↥(Bᶜ) => μ))= |
| (fun a => h a*(∫ r,F (glueAmplitude B a r) ∂Measure.pi (fun _ : ↥(Bᶜ) => μ))) := by |
| funext a |
| simp only [glueAmplitude_in] |
| exact integral_const_mul _ _ |
| rw [he] at hs |
| rw [← hs] |
| exact ht h hh hn hm |
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| end ClassicalON |
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| end |
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| end OAI |
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