File size: 3,601 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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | import OAI.Analysis.CircleDomains.Selection.SparseBaseCodes
import Mathlib.MeasureTheory.Measure.FiniteMeasure
namespace OAI
/-! The actual finite coding-measure family with a prescribed null
residual-image constraint, as a standard Borel space of codes. -/
open Set MeasureTheory
open scoped ENNReal
noncomputable section
namespace CircleDomainRigidity
variable {X Y D : Type*} [MetricSpace X] [MetricSpace Y] [TopologicalSpace D]
[MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y]
[MeasurableSpace D] [BorelSpace D]
[MeasurableSpace C(PathTime, X)] [BorelSpace C(PathTime, X)]
[MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)]
[SecondCountableTopology X] [Countable D]
abbrev ConstrainedCostCode (d : PathCodeData X Y) (p : D → X)
(c : D → ℝ≥0∞) (ν : Measure X) :=
{Γ : FiniteCostCode d p c // ν (residualImage d.dust Γ.val.val) = 0}
omit [BorelSpace D] [MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)]
[Countable D] in
theorem measurableSet_constrainedCostCode (d : PathCodeData X Y) (p : D → X)
(c : D → ℝ≥0∞) (ν : Measure X) [SFinite ν] (hB : IsClosed d.dust) :
MeasurableSet {Γ : FiniteCostCode d p c | ν (residualImage d.dust Γ.val.val) = 0} :=
(measurableSet_residualImageMass_zero hB ν).preimage
(measurable_subtype_coe.comp measurable_subtype_coe)
omit [BorelSpace D] [MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)]
[Countable D] in
theorem standardBorel_constrainedCostCode (d : PathCodeData X Y) (p : D → X)
(c : D → ℝ≥0∞) (ν : Measure X) [SFinite ν] (hB : IsClosed d.dust)
[StandardBorelSpace (FiniteCostCode d p c)] :
StandardBorelSpace (ConstrainedCostCode d p c ν) :=
(measurableSet_constrainedCostCode d p c ν hB).standardBorel
def finiteCodeMeasure (d : PathCodeData X Y) (p : D → X) (c : D → ℝ≥0∞)
(Γ : FiniteCostCode d p c) : FiniteMeasure (PathCostSpace Y D) :=
⟨pathCostMeasure d p c Γ.val, inferInstance⟩
theorem measurable_finiteCodeMeasure (d : PathCodeData X Y) (p : D → X)
(c : D → ℝ≥0∞) (hU : IsOpen d.quotientDomain) :
Measurable (finiteCodeMeasure d p c) :=
((measurable_pathCostMeasure d hU p c).comp measurable_subtype_coe).subtype_mk
def constrainedCodeMeasure (d : PathCodeData X Y) (p : D → X) (c : D → ℝ≥0∞)
(ν : Measure X) (Γ : ConstrainedCostCode d p c ν) : FiniteMeasure (PathCostSpace Y D) :=
finiteCodeMeasure d p c Γ.val
theorem measurable_constrainedCodeMeasure (d : PathCodeData X Y) (p : D → X)
(c : D → ℝ≥0∞) (ν : Measure X) (hU : IsOpen d.quotientDomain) :
Measurable (constrainedCodeMeasure d p c ν) :=
(measurable_finiteCodeMeasure d p c hU).comp measurable_subtype_coe
def sparseConstrainedCostCode (d : PathCodeData X Y) (p : D → X) (c : D → ℝ≥0∞)
(ν : Measure X) (e : ℕ ↪ ℕ) (Γ : ConstrainedCostCode d p c ν) :
ConstrainedCostCode d p c ν :=
⟨sparseFiniteCostCode d p c e Γ.val, by
change ν (residualImage d.dust (sparseTaggedPath e Γ.val.val.val)) = 0
rw [residualImage_sparse]
exact Γ.property⟩
omit [BorelSpace X] [MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)]
[SecondCountableTopology X] [Countable D] in
theorem measurable_sparseConstrainedCostCode (d : PathCodeData X Y) (p : D → X)
(c : D → ℝ≥0∞) (ν : Measure X) (e : ℕ ↪ ℕ) :
Measurable (sparseConstrainedCostCode d p c ν e) :=
((measurable_sparseFiniteCostCode d p c e).comp measurable_subtype_coe).subtype_mk
end CircleDomainRigidity
end
end OAI
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