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9655094 | 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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 | import OAI.Analysis.CircleDomains.Sobolev.BoundedUpperIntegral
import OAI.Analysis.CircleDomains.Selection.FiniteConvexSplit
import Mathlib.MeasureTheory.Measure.Prokhorov
namespace OAI
/-! A finite mixture uniformly controls all compact test configurations if every
probability distribution on configurations admits a good candidate. This is the
finite splitting step of the manuscript's fusion argument. -/
open Set MeasureTheory
open scoped Topology ENNReal
noncomputable section
namespace CircleDomainRigidity
def probabilityConvexCombination {T : Type*} [MeasurableSpace T]
(μ ν : ProbabilityMeasure T) (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b)
(hab : a + b = 1) : ProbabilityMeasure T :=
⟨ENNReal.ofReal a • (μ : Measure T) + ENNReal.ofReal b • (ν : Measure T), by
constructor
simp only [Measure.add_apply, Measure.smul_apply, measure_univ, smul_eq_mul, mul_one]
rw [← ENNReal.ofReal_add ha hb, hab, ENNReal.ofReal_one]⟩
theorem integral_probabilityConvexCombination {T : Type*} [MeasurableSpace T]
(μ ν : ProbabilityMeasure T) (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b)
(hab : a + b = 1) {f : T → ℝ}
(hμ : Integrable f (μ : Measure T)) (hν : Integrable f (ν : Measure T)) :
∫ t, f t ∂(probabilityConvexCombination μ ν a b ha hb hab : Measure T) =
a * (∫ t, f t ∂(μ : Measure T)) + b * ∫ t, f t ∂(ν : Measure T) := by
change ∫ t, f t ∂(ENNReal.ofReal a • (μ : Measure T) +
ENNReal.ofReal b • (ν : Measure T)) = _
rw [integral_add_measure (hμ.smul_measure ENNReal.ofReal_ne_top)
(hν.smul_measure ENNReal.ofReal_ne_top), integral_smul_measure, integral_smul_measure]
simp only [ENNReal.toReal_ofReal ha, ENNReal.toReal_ofReal hb, smul_eq_mul]
theorem convex_integral_vectors {I T : Type*} [MeasurableSpace T]
(f : I → T → ℝ) (hf : ∀ i (μ : ProbabilityMeasure T), Integrable (f i) (μ : Measure T)) :
Convex ℝ (Set.range (fun μ : ProbabilityMeasure T => fun i => ∫ t, f i t ∂(μ : Measure T))) := by
rintro _ ⟨μ, rfl⟩ _ ⟨ν, rfl⟩ a b ha hb hab
refine ⟨probabilityConvexCombination μ ν a b ha hb hab, ?_⟩
funext i
exact integral_probabilityConvexCombination μ ν a b ha hb hab (hf i μ) (hf i ν)
theorem exists_finite_compact_law_split
{T Z : Type*} [MetricSpace T] [CompactSpace T] [Nonempty T]
[SecondCountableTopology T] [MeasurableSpace T] [BorelSpace T]
(f : Z → T → ℝ) (A : Set Z)
(hf : ∀ z ∈ A, UpperSemicontinuous (f z))
(hf0 : ∀ z ∈ A, ∀ t, 0 ≤ f z t)
{M : ℝ} (hM : 0 < M) (hfM : ∀ z ∈ A, ∀ t, f z t ≤ M)
{b ε : ℝ} (hε : 0 < ε)
(htest : ∀ μ : ProbabilityMeasure T, ∃ z ∈ A, ∫ t, f z t ∂(μ : Measure T) ≤ b) :
∃ (n : ℕ) (z : Fin n → Z) (w : Fin n → ℝ),
(∀ i, z i ∈ A) ∧ (∀ i, 0 ≤ w i) ∧ (∑ i, w i) = 1 ∧
∀ t, ∑ i, w i * f (z i) t ≤ b + ε := by
classical
let U : A → Set (ProbabilityMeasure T) := fun z =>
{μ | (∫ t, f z.val t ∂(μ : Measure T)) < b + ε}
have hU (z : A) : IsOpen (U z) :=
(upperSemicontinuous_integral_bounded (hf z.val z.property)
(hf0 z.val z.property) hM (hfM z.val z.property)).isOpen_preimage _
have hcover : (univ : Set (ProbabilityMeasure T)) ⊆ ⋃ z : A, U z := by
intro μ _
obtain ⟨z, hz, hzt⟩ := htest μ
exact mem_iUnion.mpr ⟨⟨z, hz⟩, lt_of_le_of_lt hzt (by linarith)⟩
obtain ⟨F, hF⟩ := isCompact_univ.elim_finite_subcover U hU hcover
have hFne : F.Nonempty := by
by_contra h
have hm := hF (mem_univ (diracProba (Classical.arbitrary T)))
simp [Finset.not_nonempty_iff_eq_empty.mp h] at hm
let I := {z : A // z ∈ F}
let : Fintype I := inferInstanceAs (Fintype F)
let : Nonempty I := ⟨⟨hFne.choose, hFne.choose_spec⟩⟩
let g : I → T → ℝ := fun i => f i.val.val
have hg (i : I) (μ : ProbabilityMeasure T) : Integrable (g i) (μ : Measure T) := by
apply Integrable.of_bound (hf i.val.val i.val.property).measurable.aestronglyMeasurable M
exact Filter.Eventually.of_forall fun t => by
rw [Real.norm_eq_abs, abs_of_nonneg (hf0 _ i.val.property t)]
exact hfM _ i.val.property t
let V : Set (I → ℝ) := Set.range (fun μ : ProbabilityMeasure T =>
fun i => ∫ t, g i t ∂(μ : Measure T))
have hV : Convex ℝ V := convex_integral_vectors g hg
have hVne : V.Nonempty := ⟨_, Set.mem_range_self (diracProba (Classical.arbitrary T))⟩
have hmiss : ∀ v ∈ V, ∃ i, v i ≤ b + ε := by
rintro v ⟨μ, rfl⟩
obtain ⟨z, hz, hzμ⟩ := mem_iUnion₂.mp (hF (mem_univ μ))
exact ⟨⟨z, hz⟩, le_of_lt hzμ⟩
obtain ⟨w, hw, hw1, hwV⟩ := finite_nonnegative_separator hV hVne (b + ε) hmiss
have hpoints (t : T) : ∑ i, w i * f i.val.val t ≤ b + ε := by
have h := hwV _ (Set.mem_range_self (diracProba t))
convert h using 1
apply Finset.sum_congr rfl
intro i _
congr 1
exact (integral_dirac' (g i) t (hf i.val.val i.val.property).measurable.stronglyMeasurable).symm
let e : Fin (Fintype.card I) ≃ I := (Fintype.equivFin I).symm
refine ⟨Fintype.card I, fun i => (e i).val.val, fun i => w (e i),
fun i => (e i).val.property, fun i => hw (e i), ?_, ?_⟩
· exact (e.sum_comp w).trans hw1
· intro t
exact (e.sum_comp (fun i => w i * f i.val.val t)).le.trans (hpoints t)
end CircleDomainRigidity
end
end OAI
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