openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Approximation /PositiveIntegralBound.lean
| import OAI.MathematicalPhysics.Boltzmann.Approximation.MildCharacteristics | |
| namespace OAI | |
| section | |
| /-! # Scalar characteristic representatives for a.e. mild solutions -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| namespace BoltzmannNonuniqueness.PositiveIntegral | |
| theorem ae_path_bound_of_slices {T : ℝ} {F : ℝ × JetPrototype.Phase → ℝ} | |
| {A : ℝ → ℝ} (hFm : Measurable F) (hAm : Measurable A) | |
| (hbound : ∀ t ∈ Icc 0 T, ∀ᵐ p ∂FlightMeasure.phaseVolume, F (t,p) ≤ A t) : | |
| ∀ᵐ p ∂FlightMeasure.phaseVolume, ∀ᵐ t ∂volume.restrict (Icc (0:ℝ) T), | |
| F (t,JetBarrier.flight t p) ≤ A t := by | |
| have hm : Measurable (fun z : ℝ × JetPrototype.Phase => | |
| F (z.1,JetBarrier.flight z.1 z.2)) := | |
| hFm.comp (measurable_fst.prodMk GaussianLocal.measurable_flight) | |
| have hall : ∀ᵐ z ∂(volume.restrict (Icc (0:ℝ) T)).prod FlightMeasure.phaseVolume, | |
| F (z.1,JetBarrier.flight z.1 z.2) ≤ A z.1 := by | |
| apply (Measure.ae_prod_iff_ae_ae (measurableSet_le hm (hAm.comp measurable_fst))).mpr | |
| filter_upwards [ae_restrict_mem measurableSet_Icc] with t ht | |
| exact (FlightMeasure.flight_measurePreserving t).quasiMeasurePreserving.ae (hbound t ht) | |
| exact Measure.ae_ae_of_ae_prod | |
| ((Measure.measurePreserving_swap (μ := FlightMeasure.phaseVolume) | |
| (ν := volume.restrict (Icc 0 T))).quasiMeasurePreserving.ae hall) | |
| theorem ae_characteristic_path {T t L : ℝ} (ht : t ∈ Icc 0 T) | |
| {f₀ : JetPrototype.Phase → ℝ} (hf₀ : Measurable f₀) | |
| {F g : ℝ × JetPrototype.Phase → ℝ} (hFm : Measurable F) (hgm : Measurable g) | |
| (hmild : ∀ s ∈ Icc 0 T, (fun p => F (s,p)) =ᵐ[FlightMeasure.phaseVolume] | |
| (fun p => value T L JetBarrier.flight f₀ g (s,p))) : | |
| ∀ᵐ p ∂FlightMeasure.phaseVolume, | |
| ((fun r => F (r,JetBarrier.flight r p)) =ᵐ[volume.restrict (Icc (0:ℝ) t)] | |
| (fun r => TransportPrimitive.damped L (f₀ p) | |
| (fun s => g (s,JetBarrier.flight s p)) r)) ∧ | |
| F (t,JetBarrier.flight t p) = TransportPrimitive.damped L (f₀ p) | |
| (fun s => g (s,JetBarrier.flight s p)) t := by | |
| let A : ℝ × JetPrototype.Phase → ℝ := fun z => F (z.1,JetBarrier.flight z.1 z.2) | |
| let B : ℝ × JetPrototype.Phase → ℝ := fun z => | |
| value T L JetBarrier.flight f₀ g (z.1,JetBarrier.flight z.1 z.2) | |
| have hflow : Measurable (fun z : ℝ × JetPrototype.Phase => | |
| (z.1,JetBarrier.flight z.1 z.2)) := | |
| measurable_fst.prodMk GaussianLocal.measurable_flight | |
| have hAm : Measurable A := hFm.comp hflow | |
| have hBm : Measurable B := (measurable_value (ht.1.trans ht.2) | |
| GaussianLocal.measurable_flight hf₀ hgm).comp hflow | |
| have hslice (r : ℝ) (hr : r ∈ Icc 0 T) : | |
| (fun p => A (r,p)) =ᵐ[FlightMeasure.phaseVolume] (fun p => B (r,p)) := | |
| (FlightMeasure.flight_measurePreserving r).quasiMeasurePreserving.ae (hmild r hr) | |
| have hall : A =ᵐ[(volume.restrict (Icc 0 t)).prod FlightMeasure.phaseVolume] B := by | |
| apply (Measure.ae_prod_iff_ae_ae (measurableSet_eq_fun hAm hBm)).mpr | |
| filter_upwards [ae_restrict_mem measurableSet_Icc] with r hr | |
| exact hslice r ⟨hr.1,hr.2.trans ht.2⟩ | |
| have hp := Measure.ae_ae_of_ae_prod | |
| ((Measure.measurePreserving_swap (μ := FlightMeasure.phaseVolume) | |
| (ν := volume.restrict (Icc 0 t))).quasiMeasurePreserving.ae hall) | |
| filter_upwards [hp,hslice t ht] with p hpe hend | |
| constructor | |
| · filter_upwards [hpe,ae_restrict_mem measurableSet_Icc] with r hr hrmem | |
| exact hr.trans (value_along_flight ⟨hrmem.1,hrmem.2.trans ht.2⟩ L f₀ g p) | |
| · exact hend.trans (value_along_flight ht L f₀ g p) | |
| end BoltzmannNonuniqueness.PositiveIntegral | |
| end | |
| end | |
| section | |
| /-! # Positive upper bounds for integrable signed characteristic sources -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open scoped ENNReal | |
| namespace BoltzmannNonuniqueness.PositiveIntegral | |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} | |
| theorem ofReal_integral_le {f : X → ℝ} (hf : Integrable f μ) : | |
| ENNReal.ofReal (∫ x, f x ∂μ) ≤ ∫⁻ x, ENNReal.ofReal (f x) ∂μ := by | |
| have hm : Integrable (fun x => max (f x) 0) μ := hf.sup (integrable_zero X ℝ μ) | |
| calc | |
| _ ≤ ENNReal.ofReal (∫ x, max (f x) 0 ∂μ) := | |
| ENNReal.ofReal_le_ofReal (integral_mono hf hm (fun x => le_max_left _ _)) | |
| _ = ∫⁻ x, ENNReal.ofReal (max (f x) 0) ∂μ := | |
| ofReal_integral_eq_lintegral_ofReal hm (ae_of_all _ (fun x => le_max_right _ _)) | |
| _ = _ := by | |
| apply lintegral_congr | |
| intro x | |
| rw [ENNReal.ofReal_max,ENNReal.ofReal_zero] | |
| exact max_eq_left bot_le | |
| theorem ofReal_add_intervalIntegral_le {a b c : ℝ} (hab : a ≤ b) {g : ℝ → ℝ} | |
| (hg : IntervalIntegrable g volume a b) : | |
| ENNReal.ofReal (c+∫ s in a..b, g s) ≤ | |
| ENNReal.ofReal c+∫⁻ s in Ioc a b, ENNReal.ofReal (g s) := by | |
| apply ENNReal.ofReal_add_le.trans | |
| apply add_le_add le_rfl | |
| rw [intervalIntegral.integral_of_le hab] | |
| exact ofReal_integral_le hg.1 | |
| end BoltzmannNonuniqueness.PositiveIntegral | |
| end | |
| end | |
| end OAI | |