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8e4d533 | 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 | import Mathlib
import OAI.Geometry.CAT0Fillings.Currents.ComponentMass
import OAI.Geometry.CAT0Fillings.Currents.SumControls
namespace OAI
section
section
open Filter Set
open Set Filter MeasureTheory TopologicalSpace
open scoped Topology ENNReal
open Set MeasureTheory
open scoped RealInnerProductSpace
open Matrix
open scoped RealInnerProductSpace MatrixOrder
open Set Filter MeasureTheory
open MeasureTheory Filter Set Metric
open scoped Topology Pointwise NNReal
open Set MeasureTheory Measure Filter Module
open Set Filter MeasureTheory Measure ContinuousLinearMap
open scoped Topology Convolution NNReal
open Set Filter MeasureTheory Measure Metric
open scoped Topology ContDiff
open Set Filter Metric
open scoped Topology NNReal
open Set MeasureTheory Filter
open scoped Topology ENNReal NNReal
namespace CAT0Fillings
open Set MeasureTheory Filter
open scoped Topology ENNReal NNReal
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] {k : ℕ}
lemma IsMetricCurrent.control_bound_uniform {T : Functional X k} (hT : IsMetricCurrent T)
{μ : Measure X} [IsFiniteMeasure μ] (hμ : Controls T μ)
{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)*μ.real univ := by
have hi : 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 hi (integrable_const M) hM
calc
_ ≤ (∏ i, (K i : ℝ)) * ∫ x, |b x| ∂μ := hT.mass_bound hμ 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)
_ = _ := by ring
lemma summable_controlled_actions {T : ℕ → Functional X k}
(hT : ∀ i, IsMetricCurrent (T i)) {μ : ℕ → Measure X}
[∀ i, IsFiniteMeasure (μ i)] (hμ : ∀ i, Controls (T i) (μ i))
(hS : Summable (fun i => (μ i).real univ))
(b : X → ℝ) (π : Fin k → X → ℝ) : Summable (fun i => T i b π) := by
by_cases h : Admissible b π
· choose K hK using h.2
obtain ⟨M,hM⟩ := h.1.2
exact (hS.mul_left ((∏ j, (K j : ℝ))*M)).of_norm_bounded
(fun i => by simpa only [Real.norm_eq_abs] using
(hT i).control_bound_uniform (hμ i) h.1 K hK hM)
· simp only [fun i => (hT i).offDomain b π h,summable_zero]
lemma isMetricCurrent_tsum_controlled {T : ℕ → Functional X k}
(hT : ∀ i, IsMetricCurrent (T i)) {μ : ℕ → Measure X}
[∀ i, IsFiniteMeasure (μ i)] (hμ : ∀ i, Controls (T i) (μ i))
(hS : Summable (fun i => (μ i).real univ)) :
IsMetricCurrent (fun b π => ∑' i, T i b π) := by
have hs := summable_controlled_actions hT hμ hS
refine ⟨?_,?_,?_,?_,?_,?_⟩
· intro b π hab
simp only [fun i => (hT i).offDomain b π hab,tsum_zero]
· intro b c π a d hb hc hπ
simp only [fun i => (hT i).linearFirst b c π a d hb hc hπ]
rw [((hs b π).mul_left a).tsum_add ((hs c π).mul_left d)]
simp only [tsum_mul_left]
· intro b π j f a c hab hf
simp only [fun i => (hT i).linearCoord b π j f a c hab hf]
rw [((hs b π).mul_left a).tsum_add ((hs b (Function.update π j f)).mul_left c)]
simp only [tsum_mul_left]
· intro b π πs hb hπs hlim
choose K hK using hπs
obtain ⟨M,hM⟩ := hb.2
apply tendsto_tsum_of_dominated_convergence
(hS.mul_left ((∏ j, (K j : ℝ))*M))
(fun i => (hT i).sequentialContinuity b π πs hb (fun j => ⟨K j,hK j⟩) hlim)
exact Eventually.of_forall (fun j i =>
(hT i).control_bound_uniform (hμ i) hb K (fun l => hK l j) hM)
· intro b π hab hloc
simp only [fun i => (hT i).locality b π hab hloc,tsum_zero]
· let := finite_sum_measure_of_summable_total μ hS
exact ⟨Measure.sum μ,inferInstance,controls_sum_general hμ (fun _ _ => rfl)⟩
end CAT0Fillings
end
end
end OAI
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