File size: 1,500 Bytes
a89d282 | 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 | 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
|