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d4317d7 | 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 | 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
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