| import Mathlib |
| import OAI.Probability.SKGap.Localization.ScalarEmpirical |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open MeasureTheory ProbabilityTheory Real Set |
| open scoped ENNReal |
|
|
| lemma weighted_four_region_bound {α : Type*} [MeasurableSpace α] (μ : Measure α) |
| (f g : α → ℝ≥0∞) (hg : Measurable g) (B F D : Set α) |
| (hB : MeasurableSet B) (hF : MeasurableSet F) (hD : MeasurableSet D) |
| (hfg : ∀ x, f x ≤ g x) (k : ℝ≥0∞) |
| (hgood : ∀ x ∈ B ∩ Fᶜ ∩ Dᶜ, f x ≤ g x*k) : |
| (∫⁻ x, f x ∂μ) ≤ (∫⁻ x in Bᶜ,g x ∂μ)+(∫⁻ x in B ∩ F,g x ∂μ)+ |
| (∫⁻ x in D,g x ∂μ)+(∫⁻ x,g x ∂μ)*k := by |
| have bound (E : Set α) : (∫⁻ x in E,f x ∂μ) ≤ (∫⁻ x in E,g x ∂μ) := lintegral_mono hfg |
| have sub (E E' : Set α) (h : E ⊆ E') : (∫⁻ x in E,g x ∂μ) ≤ (∫⁻ x in E',g x ∂μ) := |
| lintegral_mono_set h |
| have hgood' : (∫⁻ x in B ∩ Fᶜ ∩ Dᶜ,f x ∂μ) ≤ (∫⁻ x,g x ∂μ)*k := by |
| calc |
| _ ≤ ∫⁻ x in B ∩ Fᶜ ∩ Dᶜ,g x*k ∂μ := |
| lintegral_mono_ae (ae_restrict_of_forall_mem ((hB.inter hF.compl).inter hD.compl) hgood) |
| _ = (∫⁻ x in B ∩ Fᶜ ∩ Dᶜ,g x ∂μ)*k := lintegral_mul_const _ hg |
| _ ≤ (∫⁻ x,g x ∂μ)*k := mul_le_mul' (lintegral_mono' Measure.restrict_le_self le_rfl) le_rfl |
| have h3 := (lintegral_split_le μ f (B ∩ Fᶜ) D).trans |
| (add_le_add ((bound _).trans (sub _ _ (fun _ h=>h.2))) hgood') |
| have h2 := (lintegral_split_le μ f B F).trans (add_le_add (bound _) h3) |
| have h1 := (lintegral_split_le μ f univ B).trans (add_le_add |
| (by simpa only [univ_inter] using h2) (by simpa only [univ_inter] using bound Bᶜ)) |
| simpa only [Measure.restrict_univ,add_comm,add_left_comm,add_assoc] using h1 |
|
|
| end SKGap |
| end |
| end |
|
|
| end OAI |
| |
