| 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 | |