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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.ConditionalCharacteristic
variable {Ω : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω]
def character (X : Ω → ℝ) (t : ℝ) (ω : Ω) : ℂ :=
Complex.exp (t * X ω * Complex.I)
theorem measurable_character {X : Ω → ℝ} (hX : Measurable[mΩ] X) (t : ℝ) :
Measurable (character X t) := by
unfold character
fun_prop
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]
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]
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)
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 _ _)
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⟩
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⟩
end Problem367.ConditionalCharacteristic
end
end OAI
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