| import OAI.Analysis.HenonEmden.IntegratedRecovery |
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| namespace OAI |
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| universe uE uΩ |
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| section AngularIntegration |
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| open MeasureTheory Set Metric |
| noncomputable section |
| namespace HenonLaneEmden.Angular |
| variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E] |
| (μ : Measure E) [μ.IsAddHaarMeasure] |
| open LineComponents |
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| def rayConstant (n : ℕ) : ℝ := 2^(n+1)*(n:ℝ) |
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| lemma long_component_measure_le {f : E → ℝ} {x : E} (hx : ‖x‖ ≤ 2) |
| {t l M : ℝ} (ht : 0 < t) (hl : 0 < l) (hl1 : l ≤ 1) (hM : 0 ≤ M) |
| (hU : IsOpen {z : sphere (0:E) 1 × ℝ | t < f (x+z.2 • z.1.1)}) |
| (hm : (∫⁻ z in ball (0:E) 4, ENNReal.ofReal (f z) ∂μ) ≤ ENNReal.ofReal M) : |
| μ.toSphere.real {e | ENNReal.ofReal l < componentLength |
| {z : sphere (0:E) 1 × ℝ | t < f (x+z.2 • z.1.1)} e} ≤ |
| rayConstant (Module.finrank ℝ E)*M/(t*l^(Module.finrank ℝ E)) := by |
| let n := Module.finrank ℝ E |
| have hn : 0 < (n:ℝ) := Nat.cast_pos.mpr Module.finrank_pos |
| have hh := (long_component_local_mass_le μ hx hl hl1 hU).trans |
| (mul_le_mul_right hm 2) |
| have hh' := ENNReal.toReal_mono (by finiteness) hh |
| have hf : 0 ≤ (l/2)^(Module.finrank ℝ E)/(Module.finrank ℝ E:ℝ) := by positivity |
| simp only [ENNReal.toReal_mul,ENNReal.toReal_ofReal ht.le,ENNReal.toReal_ofReal hf, |
| ENNReal.toReal_ofNat,ENNReal.toReal_ofReal hM] at hh' |
| rw [le_div_iff₀ (mul_pos ht (pow_pos hl _))] |
| unfold rayConstant |
| have he : t*((l/2)^n/(n:ℝ)) = (t*l^n)/(2^n*(n:ℝ)) := by |
| rw [div_pow]; field_simp |
| rw [he] at hh' |
| have hd : 0 < (2^n:ℝ)*(n:ℝ) := mul_pos (by positivity) hn |
| have hh'' := (div_le_iff₀ hd).mp (show μ.toSphere.real _*(t*l^n)/(2^n*(n:ℝ)) ≤ 2*M by |
| convert hh' using 1; unfold Measure.real at *; ring) |
| convert hh'' using 1; dsimp [n]; rw [pow_succ]; ring |
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| end HenonLaneEmden.Angular |
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| namespace HenonLaneEmden |
| lemma integral_le_constant_add_event {Ω : Type uΩ} [MeasurableSpace Ω] (ν : Measure Ω) |
| [IsFiniteMeasure ν] {S : Set Ω} (hS : MeasurableSet S) {F : Ω → ℝ} |
| (hF : Measurable F) {a H : ℝ} (ha : 0 ≤ a) (_hH : 0 ≤ H) |
| (hFn : ∀ z, 0 ≤ F z) (hFH : ∀ z, F z ≤ H) (hout : ∀ z ∉ S, F z ≤ a) : |
| (∫ z, F z ∂ν) ≤ a*ν.real univ+H*ν.real S := by |
| have hFi : Integrable F ν := (integrable_const H).mono' hF.aestronglyMeasurable |
| (ae_of_all _ (fun z => by rw [Real.norm_eq_abs,abs_of_nonneg (hFn z)]; exact hFH z)) |
| have hIi : Integrable (S.indicator (fun _ => H)) ν := (integrable_const H).indicator hS |
| have hh := integral_mono hFi ((integrable_const a).add hIi) (fun z => ?_) |
| · simpa [integral_add (integrable_const a) hIi,integral_indicator hS,integral_const, |
| smul_eq_mul,mul_comm] using hh |
| · change F z ≤ a + S.indicator (fun _ => H) z |
| by_cases hz : z ∈ S |
| · rw [indicator_of_mem hz]; linarith [hFH z] |
| · rw [indicator_of_notMem hz]; simpa only [add_zero] using hout z hz |
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| end HenonLaneEmden |
| end |
| end AngularIntegration |
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| end OAI |
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