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9b6e014 | 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 | import OAI.Probability.ClassicalON.CoarseCodes
namespace OAI
universe uX uY
noncomputable section
open MeasureTheory Set
open scoped Topology
namespace ClassicalON
section CompactIntegral
variable {X : Type uX} {Y : Type uY} [TopologicalSpace X] [CompactSpace X]
[MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [CompactSpace Y]
[MeasurableSpace Y] [BorelSpace Y] {μ : Measure X} [IsFiniteMeasure μ]
theorem compact_integrable {f : X → ℝ} (hf : Continuous f) : Integrable f μ :=
hf.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace f)
theorem compact_integral_pos [IsProbabilityMeasure μ] {w : X → ℝ}
(hw : Continuous w) (hp : ∀ x,0<w x) : 0<∫ x,w x ∂μ := by
apply (integral_pos_iff_support_of_nonneg (fun x => (hp x).le) (compact_integrable hw)).2
have he : Function.support w=univ := by ext x; simp [Function.mem_support,(hp x).ne']
rw [he,measure_univ]
exact zero_lt_one
omit [CompactSpace Y] [MeasurableSpace Y] [BorelSpace Y] in
theorem compact_parametric_integral [FirstCountableTopology Y] [LocallyCompactSpace Y]
(f : Y → X → ℝ) (hf : Continuous f.uncurry) : Continuous (fun y => ∫ x,f y x ∂μ) := by
simpa only [Measure.restrict_univ] using
(continuous_parametric_integral_of_continuous hf (isCompact_univ : IsCompact (univ : Set X)))
end CompactIntegral
def weightedMean {X : Type uX} [MeasurableSpace X] (μ : Measure X) (w f : X → ℝ) : ℝ :=
(∫ x,w x*f x ∂μ)/(∫ x,w x ∂μ)
def ContinuousAssociated {X : Type uX} [MeasurableSpace X] [TopologicalSpace X]
[Preorder X] (μ : Measure X) (w : X → ℝ) : Prop :=
∀ f g : X → ℝ, Continuous f → Continuous g → (∀ x,0≤f x) → (∀ x,0≤g x) →
Monotone f → Monotone g →
weightedMean μ w f * weightedMean μ w g ≤ weightedMean μ w (fun x => f x*g x)
section Weighted
variable {X : Type uX} [MeasurableSpace X] [TopologicalSpace X] [CompactSpace X]
[BorelSpace X] {μ : Measure X} [IsProbabilityMeasure μ] {w : X → ℝ}
(hw : Continuous w) (hp : ∀ x,0<w x)
include hw hp
theorem weightedMean_nonneg {f : X → ℝ} (hf : ∀ x,0≤f x) : 0≤weightedMean μ w f := by
exact div_nonneg (integral_nonneg (fun x => mul_nonneg (hp x).le (hf x)))
(compact_integral_pos hw hp).le
theorem weightedMean_mono {f g : X → ℝ} (hf : Continuous f) (hg : Continuous g)
(hfg : ∀ x,f x≤g x) : weightedMean μ w f ≤ weightedMean μ w g := by
exact div_le_div_of_nonneg_right
(integral_mono (compact_integrable (hw.mul hf)) (compact_integrable (hw.mul hg))
(fun x => mul_le_mul_of_nonneg_left (hfg x) (hp x).le)) (compact_integral_pos hw hp).le
theorem weightedMean_inequality_iff {f g : X → ℝ} :
weightedMean μ w f * weightedMean μ w g ≤ weightedMean μ w (fun x => f x*g x) ↔
(∫ x,w x*f x ∂μ)*(∫ x,w x*g x ∂μ) ≤ (∫ x,w x ∂μ)*(∫ x,w x*(f x*g x) ∂μ) := by
unfold weightedMean
have hm := compact_integral_pos (μ := μ) hw hp
rw [div_mul_div_comm,div_le_div_iff₀ (mul_pos hm hm) hm]
constructor <;> intro h <;> nlinarith
end Weighted
theorem continuousAssociated_chain {X : Type uX} [LinearOrder X] [TopologicalSpace X]
[CompactSpace X] [MeasurableSpace X] [BorelSpace X] [SecondCountableTopology X]
(μ : Measure X) [IsProbabilityMeasure μ] (w : X → ℝ)
(hw : Continuous w) (hp : ∀ x,0<w x) : ContinuousAssociated μ w := by
intro f g hf hg _ _ hfm hgm
apply (weightedMean_inequality_iff hw hp).2
have hc : ∀ x y,w x*f x*(w y*g y)+w x*g x*(w y*f y) ≤
w x*(f x*g x)*w y+w x*(w y*(f y*g y)) := by
intro x y
have hdiff : 0≤(f x-f y)*(g x-g y) := by
rcases le_total x y with h | h
· exact mul_nonneg_of_nonpos_of_nonpos (sub_nonpos.mpr (hfm h)) (sub_nonpos.mpr (hgm h))
· exact mul_nonneg (sub_nonneg.mpr (hfm h)) (sub_nonneg.mpr (hgm h))
have ht := mul_nonneg (mul_nonneg (hp x).le (hp y).le) hdiff
nlinarith
have hint := integral_mono
(compact_integrable (μ := μ.prod μ) (show Continuous (fun p : X×X =>
w p.1*f p.1*(w p.2*g p.2)+w p.1*g p.1*(w p.2*f p.2)) from by fun_prop))
(compact_integrable (μ := μ.prod μ) (show Continuous (fun p : X×X =>
w p.1*(f p.1*g p.1)*w p.2+w p.1*(w p.2*(f p.2*g p.2))) from by fun_prop))
(fun p => hc p.1 p.2)
rw [integral_add (compact_integrable (by fun_prop)) (compact_integrable (by fun_prop)),
integral_add (compact_integrable (by fun_prop)) (compact_integrable (by fun_prop))] at hint
rw [integral_prod_mul (fun x => w x*f x) (fun x => w x*g x),
integral_prod_mul (fun x => w x*g x) (fun x => w x*f x),
integral_prod_mul (fun x => w x*(f x*g x)) w,
integral_prod_mul w (fun x => w x*(f x*g x))] at hint
nlinarith
end ClassicalON
end
end OAI
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