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b317718 | 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 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
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