| import OAI.MeasureTheory.Falconer.Recursion.PointRieszPotential | |
| namespace OAI | |
| /-! # A positive compact restriction with bounded measurable potential -/ | |
| namespace Falconer | |
| open Set MeasureTheory | |
| open scoped ENNReal | |
| variable {d : ℕ} | |
| theorem positive_compact_bounded_function (μ : ProbabilityMeasure (Space d)) | |
| (F : Space d → ℝ≥0∞) (hF : Measurable F) (hfinite : ∀ᵐ x ∂(μ : Measure (Space d)), F x < ∞) : | |
| ∃ K : Set (Space d), IsCompact K ∧ 0 < (μ : Measure (Space d)) K ∧ | |
| ∃ B : ℝ, 0 < B ∧ ∀ x ∈ K, F x ≤ ENNReal.ofReal B := by | |
| let S (n : ℕ) : Set (Space d) := {x | F x ≤ ENNReal.ofReal (n+1 : ℝ)} | |
| have hS (n : ℕ) : MeasurableSet (S n) := measurableSet_le hF measurable_const | |
| have hfull : (⋃ n, S n) =ᵐ[(μ : Measure (Space d))] univ := by | |
| filter_upwards [hfinite] with x hx | |
| apply propext | |
| simp only [mem_iUnion,mem_univ,iff_true] | |
| obtain ⟨n,hn⟩ := exists_nat_gt (F x).toReal | |
| refine ⟨n,?_⟩ | |
| change F x ≤ ENNReal.ofReal (n+1 : ℝ) | |
| rw [← ENNReal.ofReal_toReal hx.ne] | |
| exact ENNReal.ofReal_le_ofReal (by linarith only [hn]) | |
| have hmass : (μ : Measure (Space d)) (⋃ n, S n) ≠ 0 := by | |
| rw [measure_congr hfull,measure_univ] | |
| exact one_ne_zero | |
| obtain ⟨n,hn⟩ := exists_measure_pos_of_not_measure_iUnion_null hmass | |
| obtain ⟨K,hKS,hK,hKm⟩ := (hS n).exists_lt_isCompact hn | |
| refine ⟨K,hK,hKm,(n:ℝ)+1,by positivity,?_⟩ | |
| intro x hx | |
| exact hKS hx | |
| @[fun_prop] theorem euclideanRieszPotential_measurable (μ : Measure (Space d)) [SFinite μ] | |
| (u : ℝ) : Measurable (euclideanRieszPotential μ u) := by | |
| apply Measurable.lintegral_prod_right | |
| exact (by fun_prop : Measurable (fun p : Space d × Space d => (edist p.1 p.2)^(-u))) | |
| theorem integrated_pulled_potential (μ : Measure (Space d)) [IsFiniteMeasure μ] | |
| {m : ℕ} (P : Space d →L[ℝ] Space m) (u : ℝ) : | |
| (∫⁻ x, euclideanRieszPotential (μ.map P) u (P x) ∂μ) = | |
| PlanarFalconer.rieszEnergyOn (μ.map P) u := by | |
| rw [← lintegral_map (euclideanRieszPotential_measurable (μ.map P) u) P.measurable, | |
| PlanarFalconer.rieszEnergyOn,lintegral_prod _ (by fun_prop)] | |
| rfl | |
| end Falconer | |
| end OAI | |