| import Mathlib | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory InformationTheory Real Set | |
| open scoped NNReal ENNReal | |
| open Filter | |
| open scoped Topology | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius NNReal | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Real Set Filter | |
| open MeasureTheory.Measure | |
| open scoped ENNReal NNReal MeasureTheory Topology | |
| open MeasureTheory | |
| noncomputable section | |
| noncomputable section | |
| open MeasureTheory Set NormedSpace | |
| open scoped Topology | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius | |
| noncomputable section | |
| open Set Real | |
| open scoped Topology | |
| noncomputable section | |
| open Matrix Set Filter | |
| open scoped Topology Matrix.Norms.Frobenius | |
| noncomputable section | |
| open Matrix NormedSpace ContinuousLinearMap | |
| open scoped Matrix.Norms.Frobenius | |
| noncomputable section | |
| open Matrix | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Real Set | |
| open scoped ENNReal NNReal | |
| namespace SKRatioGaussian | |
| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] | |
| lemma integral_abs_on_set_le {F : Ω → ℝ} (hF : MemLp F 2 μ) | |
| {s : Set Ω} (hs : MeasurableSet s) : | |
| (∫ x in s, |F x| ∂μ) ≤ sqrt (∫ x, (F x)^2 ∂μ)*sqrt (μ.real s) := by | |
| have hG : MemLp (s.indicator (fun _ : Ω => (1:ℝ))) 2 μ := | |
| memLp_indicator_const 2 hs 1 (Or.inr (measure_ne_top _ _)) | |
| have hh := integral_mul_norm_le_Lp_mul_Lq Real.HolderConjugate.two_two | |
| (by simpa using hF) (by simpa using hG) | |
| have he : (fun x => ‖F x‖*‖s.indicator (fun _ : Ω => (1:ℝ)) x‖) = | |
| s.indicator (fun x => |F x|) := by | |
| funext x | |
| by_cases hx : x ∈ s <;> simp [hx,Real.norm_eq_abs] | |
| have hg : (fun x => ‖s.indicator (fun _ : Ω => (1:ℝ)) x‖^(2:ℝ)) = | |
| s.indicator (fun _ : Ω => (1:ℝ)) := by | |
| funext x | |
| by_cases hx : x ∈ s <;> simp [hx] | |
| rw [he,hg,integral_indicator hs,integral_indicator hs] at hh | |
| simpa only [Real.rpow_two,Real.norm_eq_abs,sq_abs,← Real.sqrt_eq_rpow, | |
| integral_const,smul_eq_mul,mul_one,measureReal_restrict_apply_univ] using hh | |
| omit [IsProbabilityMeasure μ] in | |
| lemma integral_exceptional_equal_bound {F G H : Ω → ℝ} {s : Set Ω} | |
| (hs : MeasurableSet s) (hF : Integrable F μ) (hG : Integrable G μ) | |
| (hH : Integrable H μ) (he : ∀ x ∉ s, F x=G x) | |
| (hb : ∀ x ∈ s, ‖F x-G x‖ ≤ H x) : | |
| |(∫ x, F x ∂μ)-(∫ x, G x ∂μ)| ≤ ∫ x in s, H x ∂μ := by | |
| rw [← integral_sub hF hG] | |
| rw [← Real.norm_eq_abs] | |
| apply (norm_integral_le_integral_norm _).trans | |
| rw [← integral_indicator hs] | |
| apply integral_mono ((hF.sub hG).norm) (hH.indicator hs) | |
| intro x | |
| change ‖F x-G x‖ ≤ s.indicator H x | |
| by_cases hx : x∈s | |
| · simpa only [Set.indicator_of_mem hx] using hb x hx | |
| · simp only [Set.indicator_of_notMem hx,he x hx,sub_self,norm_zero,le_refl] | |
| end SKRatioGaussian | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end OAI | |