File size: 2,737 Bytes
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 | import OAI.Analysis.CircleDomains.Probability.PublishedMeasureDuality
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.ContinuousMap.Compact
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
namespace OAI
/-! Continuous test integrals on a compact family of finite measures and
the uniform positive mass needed for the compact duality construction. -/
open Set MeasureTheory
open scoped BoundedContinuousFunction ENNReal NNReal
noncomputable section
namespace CircleDomainRigidity
variable {Z : Type*} [TopologicalSpace Z] [MeasurableSpace Z] [BorelSpace Z]
lemma compact_family_positive_mass {S : Set (FiniteMeasure Z)}
(hS : IsCompact S) (hne : S.Nonempty) (hz : (0 : FiniteMeasure Z) ∉ S) :
∃ m : ℝ≥0, 0 < m ∧ ∀ v ∈ S, m ≤ v.mass := by
obtain ⟨v,hv,hmin⟩ := hS.exists_isMinOn hne FiniteMeasure.continuous_mass.continuousOn
refine ⟨v.mass,?_,hmin⟩
exact pos_iff_ne_zero.mpr ((FiniteMeasure.mass_nonzero_iff v).mpr (fun he => hz (he ▸ hv)))
/-- The actual averaging operator used in compact-family separation. -/
def compactMeasureTestOperator (S : Set (FiniteMeasure Z)) :
(Z →ᵇ ℝ) →ₗ[ℝ] C(S,ℝ) where
toFun f := ⟨fun v => ∫ z, f z ∂(v.val : Measure Z),
(FiniteMeasure.continuous_integral_boundedContinuousFunction f).comp continuous_subtype_val⟩
map_add' f g := by
ext v
exact integral_add (f.integrable (μ := (v.val : Measure Z)))
(g.integrable (μ := (v.val : Measure Z)))
map_smul' r f := by
ext v
exact integral_smul r f
lemma compactMeasureTestOperator_const (S : Set (FiniteMeasure Z)) (r : ℝ) (v : S) :
compactMeasureTestOperator S (BoundedContinuousFunction.const Z r) v =
r * (v.val.mass : ℝ) := by
simp [compactMeasureTestOperator,integral_const,measureReal_def,
← FiniteMeasure.ennreal_mass,mul_comm]
lemma exists_compact_measure_constant_majorant {S : Set (FiniteMeasure Z)}
(hS : IsCompact S) (hne : S.Nonempty) (hz : (0 : FiniteMeasure Z) ∉ S)
[CompactSpace S] (u : C(S,ℝ)) :
∃ r : ℝ, 0 ≤ r ∧ ∀ v : S,
u v ≤ compactMeasureTestOperator S (BoundedContinuousFunction.const Z r) v := by
obtain ⟨m,hm,hmass⟩ := compact_family_positive_mass hS hne hz
have hm' : 0 < (m : ℝ) := hm
refine ⟨‖u‖ / m,div_nonneg (norm_nonneg _) hm'.le,?_⟩
intro v
rw [compactMeasureTestOperator_const]
have hv : (m : ℝ) ≤ v.val.mass := hmass v.val v.property
calc
u v ≤ ‖u‖ := u.apply_le_norm v
_ = (‖u‖ / m) * m := (div_mul_cancel₀ _ hm'.ne').symm
_ ≤ (‖u‖ / m) * (v.val.mass : ℝ) :=
mul_le_mul_of_nonneg_left hv (div_nonneg (norm_nonneg _) hm'.le)
end CircleDomainRigidity
end
end OAI
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