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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 | 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
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