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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367
namespace DependentWeight
variable {Ω : Type*} [MeasurableSpace Ω]
variable {μ : Measure Ω} [IsFiniteMeasure μ]
theorem integral_mul_le_of_exceptional_set
{z g : Ω → ℝ} {s : Set Ω} {a M : ℝ}
(hs : MeasurableSet s) (hz : Integrable z μ)
(hzg : Integrable (fun x => z x * g x) μ)
(ha : 0 ≤ a)
(hz_nonneg : ∀ᵐ x ∂μ, 0 ≤ z x)
(hz_bound : ∀ᵐ x ∂μ, z x ≤ M)
(hg_bound : ∀ᵐ x ∂μ, g x ≤ 1)
(hg_good : ∀ᵐ x ∂μ, x ∉ s → g x ≤ a) :
(∫ x, z x * g x ∂μ) ≤ a * (∫ x, z x ∂μ) + M * μ.real s := by
have hind : Integrable (s.indicator (fun _ : Ω => M)) μ :=
(integrable_const M).indicator hs
have hpoint : (fun x => z x * g x) ≤ᵐ[μ]
(fun x => a * z x + s.indicator (fun _ : Ω => M) x) := by
filter_upwards [hz_nonneg, hz_bound, hg_bound, hg_good] with x hz0 hzM hg1 hga
by_cases hxs : x ∈ s
· simp only [Set.indicator_of_mem hxs]
have hprod : z x * g x ≤ M :=
(mul_le_mul_of_nonneg_left hg1 hz0).trans (by simpa using hzM)
exact hprod.trans (le_add_of_nonneg_left (mul_nonneg ha hz0))
· simp only [Set.indicator_of_notMem hxs, add_zero]
simpa [mul_comm] using mul_le_mul_of_nonneg_left (hga hxs) hz0
calc
(∫ x, z x * g x ∂μ) ≤
∫ x, a * z x + s.indicator (fun _ : Ω => M) x ∂μ :=
integral_mono_ae hzg ((hz.const_mul a).add hind) hpoint
_ = a * (∫ x, z x ∂μ) + M * μ.real s := by
rw [integral_add (hz.const_mul a) hind, integral_const_mul,
integral_indicator_const M hs]
simp [smul_eq_mul, mul_comm]
theorem integral_mul_le_of_exceptional_probability
{z g : Ω → ℝ} {s : Set Ω} {a M ε : ℝ}
(hs : MeasurableSet s) (hz : Integrable z μ)
(hzg : Integrable (fun x => z x * g x) μ)
(ha : 0 ≤ a) (hM : 0 ≤ M)
(hz_nonneg : ∀ᵐ x ∂μ, 0 ≤ z x)
(hz_bound : ∀ᵐ x ∂μ, z x ≤ M)
(hg_bound : ∀ᵐ x ∂μ, g x ≤ 1)
(hg_good : ∀ᵐ x ∂μ, x ∉ s → g x ≤ a)
(hprob : μ.real s ≤ ε) :
(∫ x, z x * g x ∂μ) ≤ a * (∫ x, z x ∂μ) + M * ε := by
exact (integral_mul_le_of_exceptional_set hs hz hzg ha hz_nonneg hz_bound
hg_bound hg_good).trans (add_le_add le_rfl (mul_le_mul_of_nonneg_left hprob hM))
theorem integral_mul_le_of_tail_probability
{z g : Ω → ℝ} {a M ε : ℝ}
(hz : Integrable z μ) (hg : Measurable g)
(ha : 0 ≤ a) (hM : 0 ≤ M)
(hz_nonneg : ∀ᵐ x ∂μ, 0 ≤ z x)
(hz_bound : ∀ᵐ x ∂μ, z x ≤ M)
(hg_nonneg : ∀ᵐ x ∂μ, 0 ≤ g x)
(hg_bound : ∀ᵐ x ∂μ, g x ≤ 1)
(hprob : μ.real {x | a < g x} ≤ ε) :
(∫ x, z x * g x ∂μ) ≤ a * (∫ x, z x ∂μ) + M * ε := by
have hnorm : ∀ᵐ x ∂μ, ‖g x‖ ≤ (1 : ℝ) := by
filter_upwards [hg_nonneg, hg_bound] with x hx0 hx1
simpa [Real.norm_eq_abs, abs_of_nonneg hx0] using hx1
refine integral_mul_le_of_exceptional_probability
(measurableSet_lt measurable_const hg) hz
(hz.mul_bdd hg.aestronglyMeasurable hnorm) ha hM
hz_nonneg hz_bound hg_bound ?_ hprob
exact Filter.Eventually.of_forall (fun x hx => le_of_not_gt hx)
theorem integral_sq_mul_le_of_tail_probability
{f g : Ω → ℝ} {a M ε : ℝ}
(hf : MemLp f 2 μ) (hg : Measurable g)
(ha : 0 ≤ a) (hM : 0 ≤ M)
(hf_bound : ∀ᵐ x ∂μ, f x ^ 2 ≤ M)
(hg_nonneg : ∀ᵐ x ∂μ, 0 ≤ g x)
(hg_bound : ∀ᵐ x ∂μ, g x ≤ 1)
(hprob : μ.real {x | a < g x} ≤ ε) :
(∫ x, f x ^ 2 * g x ∂μ) ≤
a * (∫ x, f x ^ 2 ∂μ) + M * ε := by
exact integral_mul_le_of_tail_probability hf.integrable_sq hg ha hM
(Filter.Eventually.of_forall fun x => sq_nonneg (f x)) hf_bound
hg_nonneg hg_bound hprob
theorem integral_mul_le_of_dominated_tail
{z g h : Ω → ℝ} {C τ M ε : ℝ}
(hz : Integrable z μ) (hg : Measurable g)
(hC : 0 ≤ C) (hτ : 0 ≤ τ) (hM : 0 ≤ M)
(hz_nonneg : ∀ᵐ x ∂μ, 0 ≤ z x)
(hz_bound : ∀ᵐ x ∂μ, z x ≤ M)
(hg_nonneg : ∀ᵐ x ∂μ, 0 ≤ g x)
(hg_bound : ∀ᵐ x ∂μ, g x ≤ 1)
(hdom : ∀ᵐ x ∂μ, g x ≤ C * h x)
(hprob : μ.real {x | τ ≤ h x} ≤ ε) :
(∫ x, z x * g x ∂μ) ≤ (C * τ) * (∫ x, z x ∂μ) + M * ε := by
have htail : μ.real {x | C * τ < g x} ≤ μ.real {x | τ ≤ h x} := by
apply ENNReal.toReal_mono (measure_ne_top μ _)
apply measure_mono_ae
filter_upwards [hdom] with x hx
intro hx_tail
by_contra hx_bad
have hh : h x ≤ τ := (lt_of_not_ge hx_bad).le
have hmul := mul_le_mul_of_nonneg_left hh hC
exact (not_lt_of_ge (hx.trans hmul)) hx_tail
exact integral_mul_le_of_tail_probability hz hg (mul_nonneg hC hτ) hM
hz_nonneg hz_bound hg_nonneg hg_bound (htail.trans hprob)
end DependentWeight
end Problem367
end
end OAI
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