| 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 | |