| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Model |
| import OAI.Geometry.CAT0Fillings.Currents.MassBounds |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
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| namespace CAT0Fillings |
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| attribute [local instance] Classical.propDecidable |
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| universe u |
|
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| end CAT0Fillings |
|
|
| open Filter Set |
| open scoped Topology NNReal |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
|
|
| namespace CAT0Fillings |
| open Set MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section SumControls |
| variable {X : Type*} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] |
| {k : ℕ} {T : Functional X k} {S : ℕ → Functional X k} |
| {μ : ℕ → Measure X} [IsFiniteMeasure (Measure.sum μ)] |
|
|
| omit [CompactSpace X] in |
| lemma controls_sum_general (hμ : ∀ i, Controls (S i) (μ i)) |
| (hT : ∀ b π, T b π = ∑' i, S i b π) : Controls T (Measure.sum μ) := by |
| intro b π hb hπ |
| have hbi : Integrable (fun x => |b x|) (Measure.sum μ) := |
| (hb.toBoundedContinuous.integrable (Measure.sum μ)).abs |
| have hs : Summable (fun i => ∫ x, |b x| ∂μ i) := by |
| simpa only [Real.norm_eq_abs,abs_abs] using hbi.summable_integral |
| have hsa : Summable (fun i => S i b π) := |
| hs.of_norm_bounded (fun i => by simpa only [Real.norm_eq_abs] using hμ i b π hb hπ) |
| rw [hT b π,integral_sum_measure hbi] |
| calc |
| |∑' i, S i b π| ≤ ∑' i, |S i b π| := by |
| simpa only [Real.norm_eq_abs] using |
| (norm_tsum_le_tsum_norm (f := fun i => S i b π) hsa.norm) |
| _ ≤ ∑' i, ∫ x, |b x| ∂μ i := hsa.abs.tsum_le_tsum (fun i => hμ i b π hb hπ) hs |
|
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| end SumControls |
|
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| end CAT0Fillings |
|
|
| namespace CAT0Fillings |
| attribute [local instance] Classical.propDecidable |
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| attribute [local instance] Classical.propDecidable |
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|
| attribute [local instance] Classical.propDecidable |
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| attribute [local instance] Classical.propDecidable |
|
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| attribute [local instance] Classical.propDecidable |
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| attribute [local instance] Classical.propDecidable |
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| attribute [local instance] Classical.propDecidable |
| open Set Metric |
| open scoped NNReal |
|
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| end CAT0Fillings |
|
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| namespace CAT0Fillings |
| open Set MeasureTheory Filter |
| open scoped Topology NNReal ENNReal |
|
|
| universe u |
| namespace CurrentOperations |
| attribute [local instance] Classical.propDecidable |
|
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| variable {X : Type u} {Y : Type*} [MetricSpace X] [MetricSpace Y] |
| [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] |
|
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| end CurrentOperations |
| end CAT0Fillings |
| end |
| end |
|
|
| end OAI |
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|
