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import OAI.Geometry.IntegralFillings.Currents.MassBounds
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace SharpIntegralFillings
attribute [local instance] Classical.propDecidable
universe u
section Foundations
variable {X : Type u} [MetricSpace X] [MeasurableSpace X]
open scoped BoundedContinuousFunction
variable [BorelSpace X]
lemma controls_of_weak_limits {ι : Type*} {l : Filter ι} [NeBot l]
{k : ℕ} {Ts : ι → Functional X k} {T : Functional X k}
{μs : ι → FiniteMeasure X} {μ : FiniteMeasure X}
(hμ : Tendsto μs l (𝓝 μ)) (hc : ∀ j, Controls (Ts j) (μs j : Measure X))
(hT : ∀ b π, Admissible b π →
Tendsto (fun j => Ts j b π) l (𝓝 (T b π))) : Controls T (μ : Measure X) := by
intro b π hb hπ
exact le_of_tendsto_of_tendsto
(Filter.Tendsto.abs (hT b π ⟨hb, fun i => ⟨1, hπ i⟩⟩))
((continuous_integral_abs_boundedLip hb).tendsto μ |>.comp hμ)
(Eventually.of_forall fun j => hc j b π hb hπ)
lemma exists_controls_of_bounded_weak_limit [CompactSpace X]
{ι : Type*} {l : Filter ι} [NeBot l] {k : ℕ}
{Ts : ι → Functional X k} {T : Functional X k} (M : ℝ≥0)
(hTs : ∀ j, IsMetricCurrent (Ts j)) (hbound : ∀ j, mass (Ts j) ≤ (M : ℝ))
(hlim : ∀ b π, Admissible b π → Tendsto (fun j => Ts j b π) l (𝓝 (T b π))) :
∃ μ : Measure X, IsFiniteMeasure μ ∧ Controls T μ ∧ μ.real univ ≤ (M : ℝ) := by
choose μj hfin hcontrol heq using fun j => (hTs j).exists_controls_mass_eq
let νs : ι → FiniteMeasure X := fun j => ⟨μj j, hfin j⟩
have hbound' : ∀ j, νs j ∈ {ν : FiniteMeasure X | ν.mass ≤ M} := by
intro j
change (νs j).mass ≤ M
apply NNReal.coe_le_coe.mp
change (μj j).real univ ≤ (M : ℝ)
rw [← heq]
exact hbound j
obtain ⟨ν, hν, hcluster⟩ :=
(isCompact_setOfPred_finiteMeasure_le_of_compactSpace X M).exists_mapClusterPt_of_frequently (l := l)
(Eventually.of_forall hbound').frequently
let l' := comap νs (𝓝 ν) ⊓ l
have hl' : NeBot l' := neBot_inf_comap_iff_map'.mpr hcluster
let := hl'
have hνs : Tendsto νs l' (𝓝 ν) := tendsto_iff_comap.mpr inf_le_left
have hctrl : Controls T (ν : Measure X) :=
controls_of_weak_limits hνs hcontrol (fun b π h => (hlim b π h).mono_left inf_le_right)
exact ⟨ν, inferInstance, hctrl, NNReal.coe_le_coe.mpr hν⟩
lemma mass_le_of_bounded_weak_limit [CompactSpace X]
{ι : Type*} {l : Filter ι} [NeBot l] {k : ℕ}
{Ts : ι → Functional X k} {T : Functional X k} (M : ℝ≥0)
(hTs : ∀ j, IsMetricCurrent (Ts j)) (hbound : ∀ j, mass (Ts j) ≤ (M : ℝ))
(hlim : ∀ b π, Admissible b π → Tendsto (fun j => Ts j b π) l (𝓝 (T b π))) :
mass T ≤ (M : ℝ) := by
obtain ⟨μ, hfin, hcontrol, hmass⟩ :=
exists_controls_of_bounded_weak_limit M hTs hbound hlim
exact (mass_le_measure hfin hcontrol).trans hmass
omit [MeasurableSpace X] [BorelSpace X] in
lemma BoundedLip.mul {b c : X → ℝ} (hb : BoundedLip b) (hc : BoundedLip c) :
BoundedLip (fun x => b x * c x) := by
obtain ⟨⟨Kb, hKb⟩, Mb, hMb⟩ := hb
obtain ⟨⟨Kc, hKc⟩, Mc, hMc⟩ := hc
let B : ℝ≥0 := ⟨max Mb 0, le_max_right _ _⟩
let C : ℝ≥0 := ⟨max Mc 0, le_max_right _ _⟩
have hB (x : X) : |b x| ≤ (B : ℝ) := (hMb x).trans (le_max_left _ _)
have hC (x : X) : |c x| ≤ (C : ℝ) := (hMc x).trans (le_max_left _ _)
refine ⟨⟨B * Kc + C * Kb, ?_⟩, (B : ℝ) * (C : ℝ), ?_⟩
· apply lipschitzWith_iff_dist_le_mul.mpr
intro x y
rw [Real.dist_eq]
calc |b x * c x - b y * c y| = |b x * (c x - c y) + (b x - b y) * c y| := by
congr 1; ring
_ ≤ |b x| * |c x - c y| + |b x - b y| * |c y| := by
simpa only [abs_mul] using abs_add_le (b x * (c x - c y)) ((b x - b y) * c y)
_ ≤ (B : ℝ) * ((Kc : ℝ) * dist x y) + ((Kb : ℝ) * dist x y) * (C : ℝ) := by
apply add_le_add
· exact mul_le_mul (hB x) (hKc.dist_le_mul x y) (abs_nonneg _) B.coe_nonneg
· exact mul_le_mul (hKb.dist_le_mul x y) (hC y) (abs_nonneg _)
(mul_nonneg Kb.coe_nonneg dist_nonneg)
_ = ((B * Kc + C * Kb : ℝ≥0) : ℝ) * dist x y := by push_cast; ring
· intro x
rw [abs_mul]
exact mul_le_mul (hB x) (hC x) (abs_nonneg _) B.coe_nonneg
lemma IsMetricCurrent.mass_bound_uniform [CompactSpace X]
{k : ℕ} {T : Functional X k} (hT : IsMetricCurrent T)
{b : X → ℝ} (hb : BoundedLip b) {π : Fin k → X → ℝ}
(K : Fin k → ℝ≥0) (hK : ∀ i, LipschitzWith (K i) (π i))
{M : ℝ} (hM : ∀ x, |b x| ≤ M) :
|T b π| ≤ ((∏ i, (K i : ℝ)) * M) * mass T := by
obtain ⟨μ, hfin, hcontrol, heq⟩ := hT.exists_controls_mass_eq
let := hfin
have hint : Integrable (fun x => |b x|) μ := (hb.toBoundedContinuous.integrable μ).abs
have hbound : (∫ x, |b x| ∂μ) ≤ μ.real univ * M := by
simpa using integral_mono hint (integrable_const M) hM
calc |T b π| ≤ (∏ i, (K i : ℝ)) * ∫ x, |b x| ∂μ := hT.mass_bound hcontrol hb K hK
_ ≤ (∏ i, (K i : ℝ)) * (μ.real univ * M) :=
mul_le_mul_of_nonneg_left hbound (Finset.prod_nonneg fun i _ => (K i).coe_nonneg)
_ = ((∏ i, (K i : ℝ)) * M) * mass T := by rw [heq]; ring
lemma IsMetricCurrent.eq_zero_of_mass_eq_zero [CompactSpace X]
{k : ℕ} {T : Functional X k} (hT : IsMetricCurrent T) (hm : mass T = 0) :
T = fun _ _ => 0 := by
funext b π
by_cases h : Admissible b π
· choose K hK using h.2
obtain ⟨M, hM⟩ := h.1.2
have hle := hT.mass_bound_uniform h.1 K hK hM
rw [hm, mul_zero] at hle
exact abs_eq_zero.mp (le_antisymm hle (abs_nonneg _))
· exact hT.offDomain b π h
end Foundations
end SharpIntegralFillings
end
end OAI
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