| import Mathlib |
| import OAI.Probability.GaussianInformation.Kernel.ReferenceCharge |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Metric Set |
| open scoped ENNReal |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| open MeasureTheory ProbabilityTheory |
| open MeasureTheory |
| open scoped ENNReal |
|
|
| namespace KernelPotential.FiniteMessage |
| open MeasureTheory ProbabilityTheory Set Function |
| open scoped ENNReal |
| variable {d N : ℕ} |
| lemma fibre_measurable (v : Fin N) : MeasurableSet {x : Ambient d × Fin N | x.2=v} := |
| measurable_snd (measurableSet_singleton v) |
| lemma sum_restrict_fibres (μ : Measure (Ambient d × Fin N)) : |
| (∑ v : Fin N,μ.restrict {x|x.2=v})=μ := by |
| have hu : (⋃ v : Fin N,{x : Ambient d × Fin N|x.2=v})=univ := by ext x; simp |
| have hd : Pairwise (Disjoint on (fun v : Fin N => {x : Ambient d × Fin N|x.2=v})) := by |
| intro v w hvw |
| change Disjoint {x : Ambient d × Fin N | x.2=v} {x|x.2=w} |
| rw [Set.disjoint_left] |
| intro x hx hxw |
| exact hvw (hx.symm.trans hxw) |
| have hh := Measure.restrict_iUnion (μ := μ) hd (fun v => fibre_measurable v) |
| rw [hu,Measure.restrict_univ] at hh |
| simpa only [Measure.sum_fintype] using hh.symm |
| lemma component_le (μ : Measure (Ambient d × Fin N)) (v : Fin N) : component μ v ≤ μ.map Prod.fst := |
| Measure.map_mono (Measure.restrict_le_self) measurable_fst |
| lemma sum_components (μ : Measure (Ambient d × Fin N)) : (∑ v,component μ v)=μ.map Prod.fst := by |
| rw [← sum_restrict_fibres μ,Measure.map_finset_sum] |
| · congr 1 |
| ext v |
| rw [sum_restrict_fibres] |
| rfl |
| · exact measurable_fst.aemeasurable |
| lemma component_finite (μ : Measure (Ambient d × Fin N)) [IsFiniteMeasure μ] (v : Fin N) : |
| IsFiniteMeasure (component μ v) := by unfold component; infer_instance |
| lemma component_apply (μ : Measure (Ambient d × Fin N)) (v : Fin N) (s : Set (Ambient d)) (hs : MeasurableSet s) : |
| component μ v s=μ ((Prod.fst ⁻¹' s) ∩ {x|x.2=v}) := by |
| rw [component,Measure.map_apply measurable_fst hs,Measure.restrict_apply (measurable_fst hs)] |
| lemma component_univ (μ : Measure (Ambient d × Fin N)) (v : Fin N) : |
| component μ v univ=μ {x|x.2=v} := by simp [component_apply μ v univ MeasurableSet.univ] |
| lemma component_sum {ι : Type*} (s : Finset ι) (μ : ι → Measure (Ambient d × Fin N)) (v : Fin N) : |
| component (∑ i ∈ s,μ i) v=∑ i ∈ s,component (μ i) v := by |
| ext t ht |
| simp only [component_apply _ _ _ ht,Measure.finsetSum_apply] |
| lemma component_graph (ρ : Measure (Ambient d)) (w v : Fin N) : |
| component (ρ.map (fun s => (s,w))) v=if w=v then ρ else 0 := by |
| ext t ht |
| rw [component_apply _ _ _ ht,Measure.map_apply (show Measurable (fun s : Ambient d => (s,w)) from measurable_id.prodMk measurable_const) |
| ((measurable_fst ht).inter (fibre_measurable v))] |
| by_cases h : w=v |
| · simp only [ite_eq_left h] |
| congr 1 |
| ext s |
| simp [h] |
| · simp [h] |
| lemma lintegral_component (μ : Measure (Ambient d × Fin N)) (v : Fin N) |
| (f : Ambient d → ℝ≥0∞) (hf : Measurable f) : |
| (∫⁻ s,f s ∂component μ v)=∫⁻ x in {x|x.2=v},f x.1 ∂μ := by |
| exact lintegral_map hf measurable_fst |
| lemma lintegral_split (μ : Measure (Ambient d × Fin N)) |
| (f : Ambient d × Fin N → ℝ≥0∞) (hf : Measurable f) : |
| (∫⁻ x,f x ∂μ)=∑ v : Fin N,∫⁻ s,f (s,v) ∂component μ v := by |
| conv_lhs => rw [← sum_restrict_fibres μ] |
| rw [lintegral_finsetSum_measure] |
| apply Finset.sum_congr rfl |
| intro v _ |
| rw [lintegral_component μ v (fun s => f (s,v)) (hf.comp (measurable_id.prodMk measurable_const))] |
| apply lintegral_congr_ae |
| filter_upwards [ae_restrict_mem (fibre_measurable v)] with x hx |
| change x.2=v at hx |
| exact congrArg f (Prod.ext rfl hx) |
| end KernelPotential.FiniteMessage |
|
|
| end OAI |
|
|
