| import OAI.Probability.InvariantIsing.Spectral.PositiveSpectralLabels |
| import OAI.Probability.InvariantIsing.Spectral.FiniteSpectralMeasure |
|
|
| namespace OAI |
|
|
| /-! The positive part of a finite random label has the same distribution. -/ |
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped BigOperators Classical |
| namespace InvariantIsing |
|
|
| variable {Ω A : Type*} [MeasurableSpace Ω] [Fintype A] |
| [MeasurableSpace A] [MeasurableSingletonClass A] |
|
|
| def finiteIndexWeight (P : Measure Ω) (g : Ω → A) (a : A) : ℝ := |
| P.real (g ⁻¹' {a}) |
|
|
| lemma finiteIndexWeight_sum (P : Measure Ω) [IsProbabilityMeasure P] |
| (g : Ω → A) (hg : Measurable g) : ∑ a, finiteIndexWeight P g a=1 := by |
| have hh := sum_measureReal_preimage_singleton (μ := P) Finset.univ |
| (f := g) (fun a _ => hg (measurableSet_singleton a)) |
| simpa only [finiteIndexWeight,Finset.coe_univ,preimage_univ,probReal_univ] using hh |
|
|
| omit [MeasurableSpace A] [MeasurableSingletonClass A] in |
| lemma ae_finiteIndexWeight_pos (P : Measure Ω) [IsProbabilityMeasure P] (g : Ω → A) : |
| ∀ᵐ x ∂P, 0 < finiteIndexWeight P g (g x) := by |
| have hi a : ∀ᵐ x ∂P, g x=a → 0 < finiteIndexWeight P g a := by |
| by_cases hp : 0 < finiteIndexWeight P g a |
| · exact Eventually.of_forall (fun _ _ => hp) |
| · have hz : P (g ⁻¹' {a})=0 := (measureReal_eq_zero_iff).mp |
| (le_antisymm (le_of_not_gt hp) measureReal_nonneg) |
| filter_upwards [compl_mem_ae_iff.mpr hz] with x hx |
| exact fun h => (hx h).elim |
| filter_upwards [ae_all_iff.mpr hi] with x hx |
| exact hx _ rfl |
|
|
| omit [MeasurableSpace A] [MeasurableSingletonClass A] in |
| lemma positiveIndexWeight (P : Measure Ω) [IsProbabilityMeasure P] |
| (g : Ω → A) (fallback : {a // 0 < finiteIndexWeight P g a}) |
| (a : {a // 0 < finiteIndexWeight P g a}) : |
| finiteIndexWeight P (positiveSpectralLabel (finiteIndexWeight P g) fallback ∘ g) a= |
| finiteIndexWeight P g a := by |
| apply measureReal_congr |
| filter_upwards [ae_finiteIndexWeight_pos P g] with x hx |
| simp only [Function.comp_apply,positiveSpectralLabel,dite_eq_left hx, |
| mem_preimage,mem_singleton_iff] |
| exact propext Subtype.ext_iff |
|
|
| lemma finiteIndex_map (P : Measure Ω) [IsProbabilityMeasure P] |
| (g : Ω → A) (hg : Measurable g) (c : A → ℝ) : |
| P.map (c ∘ g)=finiteSpectralMeasure (finiteIndexWeight P g) c := by |
| have hc : Measurable c := measurable_of_finite c |
| rw [← Measure.map_map hc hg,Measure.map_eq_sum P g hg,Measure.map_sum hc.aemeasurable] |
| unfold finiteSpectralMeasure |
| congr 1 |
| funext a |
| rw [Measure.map_smul _ hc.aemeasurable,Measure.map_dirac,finiteIndexWeight, |
| measureReal_def,ENNReal.ofReal_toReal (measure_ne_top P _)] |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
