| import Mathlib.MeasureTheory.Integral.Lebesgue.Markov |
| import Mathlib.MeasureTheory.Measure.Typeclasses.Probability |
| import Mathlib.Topology.Semicontinuity.Basic |
| import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory Filter |
| open scoped ENNReal NNReal |
|
|
| theorem exists_positive_compact_sublevel {X : Type*} [TopologicalSpace X] |
| [MeasurableSpace X] [BorelSpace X] [T2Space X] |
| (μ : Measure X) [IsProbabilityMeasure μ] {K : Set X} (hK : IsCompact K) |
| (hμ : ∀ᵐ x ∂μ, x ∈ K) (f : X → ℝ≥0∞) (hf : LowerSemicontinuous f) |
| {p : ℝ} (hp : 0 < p) (hI : (∫⁻ x, (f x)^p ∂μ) < ∞) : |
| ∃ A : Set X, IsCompact A ∧ A ⊆ K ∧ 0 < μ A ∧ |
| ∃ C : ℝ≥0∞, C ≠ ∞ ∧ ∀ x ∈ A, f x ≤ C := by |
| let A : ℕ → Set X := fun n => K ∩ f ⁻¹' Iic (n : ℝ≥0∞) |
| have hfinite : ∀ᵐ x ∂μ, f x < ∞ := by |
| filter_upwards [ae_lt_top (hf.measurable.pow_const p) hI.ne] with x hx |
| by_contra h |
| have ht : f x = ∞ := top_unique (not_lt.mp h) |
| rw [ht, ENNReal.top_rpow_of_pos hp] at hx |
| exact lt_irrefl _ hx |
| have hfull : ∀ᵐ x ∂μ, x ∈ ⋃ n, A n := by |
| filter_upwards [hμ, hfinite] with x hx hfx |
| obtain ⟨n, hn⟩ := ENNReal.exists_nat_gt hfx.ne |
| exact mem_iUnion.mpr ⟨n, hx, hn.le⟩ |
| obtain ⟨n, hn⟩ : ∃ n, 0 < μ (A n) := by |
| by_contra h |
| have hz : ∀ n, μ (A n) = 0 := fun n => le_antisymm (not_lt.mp (fun hn => h ⟨n, hn⟩)) bot_le |
| have hzero : μ (⋃ n, A n) = 0 := measure_iUnion_null hz |
| have hc : μ (⋃ n, A n)ᶜ = 0 := by |
| change μ {x | x ∉ ⋃ n, A n} = 0 |
| exact (ae_iff.mp hfull) |
| have hu : μ univ = 0 := by |
| rw [← union_compl_self (⋃ n, A n)] |
| exact measure_union_null hzero hc |
| simp at hu |
| refine ⟨A n, hK.inter_right (hf.isClosed_preimage n), inter_subset_left, hn, |
| n, ENNReal.natCast_ne_top n, fun x hx => hx.2⟩ |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
