| import OAI.Probability.IsingPerceptron.RetainedDisplacement |
| import OAI.Probability.IsingPerceptron.NamespaceSplit |
|
|
| namespace OAI |
|
|
| /-! Pushforward of an integrable terminal Gibbs tilt. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped ENNReal |
| namespace InvariantIsing |
|
|
| lemma gibbsProbability_map_normalize {X Y : Type} [MeasurableSpace X] [MeasurableSpace Y] |
| [Nonempty X] [Nonempty Y] (ν : Measure X) [IsProbabilityMeasure ν] |
| {f : X → Y} (hf : Measurable f) {H : Y → ℝ} (hH : Measurable H) |
| (hi : Integrable (fun x => Real.exp (H (f x))) ν) : |
| (gibbsProbability ν (fun x => H (f x))).map f = |
| normalizeMass ((ν.map f).withDensity (fun y => ENNReal.ofReal (Real.exp (H y)))) := by |
| have hI : Integrable (fun y => Real.exp (H y)) (ν.map f) := |
| (integrable_map_measure hH.exp.aestronglyMeasurable hf.aemeasurable).mpr hi |
| have hmass : ((ν.map f).withDensity (fun y => ENNReal.ofReal (Real.exp (H y)))) Set.univ = |
| ENNReal.ofReal (∫ y, Real.exp (H y) ∂ν.map f) := by |
| rw [withDensity_apply _ MeasurableSet.univ,setLIntegral_univ, |
| ← ofReal_integral_eq_lintegral_ofReal hI (ae_of_all _ fun y => (Real.exp_pos _).le)] |
| rw [gibbsProbability_eq_tilted _ _ hi, ← normalizeMass_exp ν (fun x => H (f x)) (hH.comp hf) hi] |
| apply normalizeMass_weight_map ν hf hH.exp.ennreal_ofReal |
| rw [hmass] |
| exact ⟨ENNReal.ofReal_pos.mpr (integral_exp_pos hI), ENNReal.ofReal_lt_top⟩ |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
