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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 99 | 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
namespace CAT0Fillings
attribute [local instance] Classical.propDecidable
universe u
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
end SumControls
end CAT0Fillings
namespace CAT0Fillings
attribute [local instance] Classical.propDecidable
attribute [local instance] Classical.propDecidable
attribute [local instance] Classical.propDecidable
attribute [local instance] Classical.propDecidable
attribute [local instance] Classical.propDecidable
attribute [local instance] Classical.propDecidable
attribute [local instance] Classical.propDecidable
open Set Metric
open scoped NNReal
end CAT0Fillings
namespace CAT0Fillings
open Set MeasureTheory Filter
open scoped Topology NNReal ENNReal
universe u
namespace CurrentOperations
attribute [local instance] Classical.propDecidable
variable {X : Type u} {Y : Type*} [MetricSpace X] [MetricSpace Y]
[MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y]
end CurrentOperations
end CAT0Fillings
end
end
end OAI
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