| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
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| namespace Problem367.ConditionalCharacteristic |
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| variable {Ω : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] |
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| def character (X : Ω → ℝ) (t : ℝ) (ω : Ω) : ℂ := |
| Complex.exp (t * X ω * Complex.I) |
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| theorem measurable_character {X : Ω → ℝ} (hX : Measurable[mΩ] X) (t : ℝ) : |
| Measurable (character X t) := by |
| unfold character |
| fun_prop |
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| theorem integrable_character {P : Measure Ω} [IsFiniteMeasure P] |
| {X : Ω → ℝ} (hX : Measurable[mΩ] X) (t : ℝ) : |
| Integrable (character X t) P := by |
| apply Integrable.mono' (integrable_const (1 : ℝ)) |
| (measurable_character hX t).aestronglyMeasurable |
| apply Filter.Eventually.of_forall |
| intro ω |
| simp [character, ← Complex.ofReal_mul] |
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| theorem map_restrict_eq_smul_of_condExp_character |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| (hm : m ≤ mΩ) |
| {X : Ω → ℝ} (hX : Measurable[mΩ] X) |
| (ν : Measure ℝ) [IsProbabilityMeasure ν] |
| (hchar : ∀ t : ℝ, P[character X t | m] =ᵐ[P] fun _ => charFun ν t) |
| {s : Set Ω} (hs : MeasurableSet[m] s) : |
| Measure.map X (P.restrict s) = P s • ν := by |
| let : IsFiniteMeasure (P s • ν) := ν.smul_finite (measure_ne_top P s) |
| apply Measure.ext_of_charFun |
| funext t |
| rw [charFun_apply_real, integral_map hX.aemeasurable (by fun_prop), |
| charFun_apply_real, integral_smul_measure] |
| change (∫ ω in s, character X t ω ∂P) = _ |
| rw [← setIntegral_condExp hm (integrable_character hX t) hs] |
| calc |
| (∫ ω in s, P[character X t | m] ω ∂P) = |
| ∫ _ in s, charFun ν t ∂P := |
| integral_congr_ae (ae_restrict_of_ae (hchar t)) |
| _ = _ := by simp [charFun_apply_real, measureReal_def] |
|
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| theorem map_eq_of_condExp_character |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| (hm : m ≤ mΩ) |
| {X : Ω → ℝ} (hX : Measurable[mΩ] X) |
| (ν : Measure ℝ) [IsProbabilityMeasure ν] |
| (hchar : ∀ t : ℝ, P[character X t | m] =ᵐ[P] fun _ => charFun ν t) : |
| Measure.map X P = ν := by |
| simpa using map_restrict_eq_smul_of_condExp_character hm hX ν hchar |
| (MeasurableSet.univ : MeasurableSet[m] Set.univ) |
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| theorem indepFun_of_condExp_character |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| (hm : m ≤ mΩ) |
| {X : Ω → ℝ} (hX : Measurable[mΩ] X) |
| (ν : Measure ℝ) [IsProbabilityMeasure ν] |
| (hchar : ∀ t : ℝ, P[character X t | m] =ᵐ[P] fun _ => charFun ν t) |
| {E : Type*} [MeasurableSpace E] {Y : Ω → E} (hY : Measurable[m] Y) : |
| IndepFun X Y P := by |
| rw [indepFun_iff_measure_inter_preimage_eq_mul] |
| intro a b ha hb |
| have hrestrict := congrArg (fun μ : Measure ℝ => μ a) |
| (map_restrict_eq_smul_of_condExp_character hm hX ν hchar (hY hb)) |
| have hlaw := map_eq_of_condExp_character hm hX ν hchar |
| rw [Measure.map_apply hX ha, Measure.restrict_apply (hX ha), |
| Measure.smul_apply, smul_eq_mul, ← hlaw, Measure.map_apply hX ha] at hrestrict |
| exact hrestrict.trans (mul_comm _ _) |
|
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| theorem hasLaw_of_condExp_character |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| (hm : m ≤ mΩ) |
| {X : Ω → ℝ} (hX : Measurable[mΩ] X) |
| (ν : Measure ℝ) [IsProbabilityMeasure ν] |
| (hchar : ∀ t : ℝ, P[character X t | m] =ᵐ[P] fun _ => charFun ν t) : |
| HasLaw X ν P := |
| ⟨hX.aemeasurable, map_eq_of_condExp_character hm hX ν hchar⟩ |
|
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| theorem gaussian_law_and_indep_of_condExp_character |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| (hm : m ≤ mΩ) |
| {X : Ω → ℝ} (hX : Measurable[mΩ] X) (v : ℝ≥0) |
| (hchar : ∀ t : ℝ, P[character X t | m] =ᵐ[P] |
| fun _ => Complex.exp (-((v : ℝ) : ℂ) * (t : ℂ) ^ 2 / 2)) |
| {E : Type*} [MeasurableSpace E] {Y : Ω → E} (hY : Measurable[m] Y) : |
| HasLaw X (gaussianReal 0 v) P ∧ IndepFun X Y P := by |
| have hc : ∀ t : ℝ, P[character X t | m] =ᵐ[P] |
| fun _ => charFun (gaussianReal 0 v) t := by |
| intro t |
| simpa only [charFun_gaussianReal, Complex.ofReal_zero, mul_zero, zero_mul, zero_sub, |
| neg_mul, neg_div] using hchar t |
| exact ⟨hasLaw_of_condExp_character hm hX _ hc, |
| indepFun_of_condExp_character hm hX _ hc hY⟩ |
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| end Problem367.ConditionalCharacteristic |
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| end |
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| end OAI |
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