| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem367 | |
| def linearSignalTilt {Ω : Type*} [MeasurableSpace Ω] | |
| (mu : Measure Ω) (q : Ω → ℝ) (a : ℝ) : Measure Ω := | |
| mu.withDensity (fun x => ENNReal.ofReal (1 + a * q x)) | |
| theorem linearSignalTilt_density_nonneg {a q : ℝ} | |
| (ha : |a| ≤ 1) (hq : |q| ≤ 1) : 0 ≤ 1 + a * q := by | |
| have hab : |a * q| ≤ 1 := by | |
| rw [abs_mul] | |
| calc | |
| |a| * |q| ≤ 1 * |q| := mul_le_mul_of_nonneg_right ha (abs_nonneg q) | |
| _ ≤ 1 := by simpa using hq | |
| have := (abs_le.mp hab).1 | |
| linarith | |
| theorem integrable_boundedSignal_mul {Ω : Type*} [MeasurableSpace Ω] | |
| {mu : Measure Ω} {q f : Ω → ℝ} (hq : Measurable q) | |
| (hqb : ∀ᵐ x ∂mu, |q x| ≤ 1) (hf : Integrable f mu) : | |
| Integrable (fun x => q x * f x) mu := by | |
| refine hf.norm.mono' (hq.aestronglyMeasurable.mul hf.aestronglyMeasurable) ?_ | |
| filter_upwards [hqb] with x hx | |
| rw [norm_mul, Real.norm_eq_abs] | |
| simpa only [one_mul] using mul_le_mul_of_nonneg_right hx (norm_nonneg (f x)) | |
| theorem integrable_linearSignalTilt {Ω : Type*} [MeasurableSpace Ω] | |
| {mu : Measure Ω} {q f : Ω → ℝ} {a : ℝ} (hq : Measurable q) | |
| (hqb : ∀ᵐ x ∂mu, |q x| ≤ 1) (ha : |a| ≤ 1) (hf : Integrable f mu) : | |
| Integrable f (linearSignalTilt mu q a) := by | |
| rw [linearSignalTilt, integrable_withDensity_iff_integrable_smul' | |
| (by fun_prop) (Filter.Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))] | |
| apply (hf.add ((integrable_boundedSignal_mul hq hqb hf).const_mul a)).congr | |
| filter_upwards [hqb] with x hx | |
| rw [ENNReal.toReal_ofReal (linearSignalTilt_density_nonneg ha hx), smul_eq_mul] | |
| simp only [Pi.add_apply] | |
| ring | |
| theorem integral_linearSignalTilt {Ω : Type*} [MeasurableSpace Ω] | |
| {mu : Measure Ω} {q f : Ω → ℝ} {a : ℝ} (hq : Measurable q) | |
| (hqb : ∀ᵐ x ∂mu, |q x| ≤ 1) (ha : |a| ≤ 1) (hf : Integrable f mu) : | |
| (∫ x, f x ∂linearSignalTilt mu q a) = | |
| (∫ x, f x ∂mu) + a * ∫ x, q x * f x ∂mu := by | |
| rw [linearSignalTilt, integral_withDensity_eq_integral_toReal_smul | |
| (by fun_prop) (Filter.Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))] | |
| have heq : (fun x => (ENNReal.ofReal (1 + a * q x)).toReal • f x) =ᵐ[mu] | |
| (fun x => f x + a * (q x * f x)) := by | |
| filter_upwards [hqb] with x hx | |
| rw [ENNReal.toReal_ofReal (linearSignalTilt_density_nonneg ha hx), smul_eq_mul] | |
| ring | |
| rw [integral_congr_ae heq, | |
| integral_add hf ((integrable_boundedSignal_mul hq hqb hf).const_mul a), | |
| integral_const_mul] | |
| theorem linearSignalTilt_probability {Ω : Type*} [MeasurableSpace Ω] | |
| {mu : Measure Ω} [IsProbabilityMeasure mu] {q : Ω → ℝ} {a : ℝ} | |
| (hq : Measurable q) (hqb : ∀ᵐ x ∂mu, |q x| ≤ 1) (ha : |a| ≤ 1) | |
| (hmean : (∫ x, q x ∂mu) = 0) : IsProbabilityMeasure (linearSignalTilt mu q a) := by | |
| have hiq : Integrable q mu := by | |
| refine (integrable_const (1 : ℝ)).mono' hq.aestronglyMeasurable ?_ | |
| simpa only [Real.norm_eq_abs] using hqb | |
| have hid : Integrable (fun x => 1 + a * q x) mu := (integrable_const 1).add (hiq.const_mul a) | |
| have hnonneg : 0 ≤ᵐ[mu] (fun x => 1 + a * q x) := | |
| hqb.mono (fun _ hx => linearSignalTilt_density_nonneg ha hx) | |
| have hint : (∫ x, 1 + a * q x ∂mu) = 1 := by | |
| rw [integral_add (integrable_const 1) (hiq.const_mul a), integral_const_mul, hmean] | |
| simp | |
| constructor | |
| rw [linearSignalTilt, withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ, | |
| ← ofReal_integral_eq_lintegral_ofReal hid hnonneg, hint] | |
| simp | |
| theorem integral_eq_zero_of_ae_sign_reversal {Ω : Type*} [MeasurableSpace Ω] | |
| {mu : Measure Ω} (T : Ω ≃ᵐ Ω) (hT : MeasurePreserving T mu mu) | |
| {f : Ω → ℝ} (hf : ∀ᵐ x ∂mu, f (T x) = -f x) : (∫ x, f x ∂mu) = 0 := by | |
| have h := hT.integral_comp' f | |
| rw [integral_congr_ae hf, integral_neg] at h | |
| linarith | |
| theorem integral_eq_zero_of_sign_reversal {Ω : Type*} [MeasurableSpace Ω] | |
| {mu : Measure Ω} (T : Ω ≃ᵐ Ω) (hT : MeasurePreserving T mu mu) | |
| {f : Ω → ℝ} (hf : ∀ x, f (T x) = -f x) : (∫ x, f x ∂mu) = 0 := by | |
| exact integral_eq_zero_of_ae_sign_reversal T hT (Filter.Eventually.of_forall hf) | |
| theorem integral_linearSignalTilt_even {Ω : Type*} [MeasurableSpace Ω] | |
| {mu : Measure Ω} {q f : Ω → ℝ} {a : ℝ} (hq : Measurable q) | |
| (hqb : ∀ᵐ x ∂mu, |q x| ≤ 1) (ha : |a| ≤ 1) (hf : Integrable f mu) | |
| (T : Ω ≃ᵐ Ω) (hT : MeasurePreserving T mu mu) | |
| (hqodd : ∀ x, q (T x) = -q x) (hfeven : ∀ x, f (T x) = f x) : | |
| (∫ x, f x ∂linearSignalTilt mu q a) = ∫ x, f x ∂mu := by | |
| rw [integral_linearSignalTilt hq hqb ha hf] | |
| have hz : (∫ x, q x * f x ∂mu) = 0 := | |
| integral_eq_zero_of_sign_reversal T hT (fun x => by rw [hqodd, hfeven, neg_mul]) | |
| rw [hz] | |
| ring | |
| theorem integral_linearSignalTilt_ae_even {Ω : Type*} [MeasurableSpace Ω] | |
| {mu : Measure Ω} {q f : Ω → ℝ} {a : ℝ} (hq : Measurable q) | |
| (hqb : ∀ᵐ x ∂mu, |q x| ≤ 1) (ha : |a| ≤ 1) (hf : Integrable f mu) | |
| (T : Ω ≃ᵐ Ω) (hT : MeasurePreserving T mu mu) | |
| (hqodd : ∀ᵐ x ∂mu, q (T x) = -q x) (hfeven : ∀ᵐ x ∂mu, f (T x) = f x) : | |
| (∫ x, f x ∂linearSignalTilt mu q a) = ∫ x, f x ∂mu := by | |
| rw [integral_linearSignalTilt hq hqb ha hf] | |
| have hz : (∫ x, q x * f x ∂mu) = 0 := by | |
| apply integral_eq_zero_of_ae_sign_reversal T hT | |
| filter_upwards [hqodd, hfeven] with x hxq hxf | |
| rw [hxq, hxf, neg_mul] | |
| rw [hz] | |
| ring | |
| theorem map_withDensity_comp {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] | |
| {mu : Measure Ω} {f : Ω → X} {w : X → ENNReal} | |
| (hf : Measurable f) (hw : Measurable w) : | |
| Measure.map f (mu.withDensity (w ∘ f)) = (Measure.map f mu).withDensity w := by | |
| ext s hs | |
| rw [Measure.map_apply hf hs, withDensity_apply _ (hf hs), withDensity_apply _ hs, | |
| setLIntegral_map hs hw hf] | |
| rfl | |
| theorem map_linearSignalTilt {Ω : Type*} [MeasurableSpace Ω] | |
| {mu : Measure Ω} {q : Ω → ℝ} (hq : Measurable q) (a : ℝ) : | |
| Measure.map q (linearSignalTilt mu q a) = linearSignalTilt (Measure.map q mu) id a := by | |
| unfold linearSignalTilt | |
| simpa only [Function.comp_def, id_eq] using | |
| (map_withDensity_comp (mu := mu) (f := q) | |
| (w := fun y : ℝ => ENNReal.ofReal (1 + a * y)) hq (by fun_prop)) | |
| end Problem367 | |
| end | |
| end OAI | |