| import OAI.Probability.FreeIsing.Spins.Chain | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem367 | |
| theorem integral_finiteLaw {Ω E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] | |
| [Fintype E] [MeasurableSingletonClass E] | |
| {mu : Measure Ω} [IsFiniteMeasure mu] {X : Ω → E} (hX : Measurable X) | |
| (weight : E → ℝ) (hweight : ∀ e, 0 ≤ weight e) | |
| (hlaw : ∀ e, mu {ω | X ω = e} = ENNReal.ofReal (weight e)) (f : E → ℝ) : | |
| ∫ ω, f (X ω) ∂mu = ∑ e : E, weight e * f e := by | |
| rw [← integral_map hX.aemeasurable (measurable_of_finite f).aestronglyMeasurable] | |
| rw [integral_fintype (Integrable.of_finite : Integrable f (Measure.map X mu))] | |
| apply Finset.sum_congr rfl | |
| intro e he | |
| rw [Measure.real, Measure.map_apply hX (measurableSet_singleton e)] | |
| change (mu {ω | X ω = e}).toReal * f e = weight e * f e | |
| rw [hlaw e, ENNReal.toReal_ofReal (hweight e)] | |
| theorem integrable_finiteLaw {Ω E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] | |
| [Fintype E] [MeasurableSingletonClass E] | |
| {mu : Measure Ω} [IsFiniteMeasure mu] {X : Ω → E} (hX : Measurable X) | |
| (f : E → ℝ) : Integrable (fun ω => f (X ω)) mu := by | |
| exact (integrable_map_measure (measurable_of_finite f).aestronglyMeasurable | |
| hX.aemeasurable).mp Integrable.of_finite | |
| local instance : MeasurableSingletonClass IsingSpin where | |
| measurableSet_singleton _ := by trivial | |
| theorem integral_finiteSpinCoordinates {d : ℕ} {I : Type*} [Fintype I] [DecidableEq I] | |
| {mu : Measure (TreeVertex d → IsingSpin)} [IsFiniteMeasure mu] | |
| (p : I → TreeVertex d) (weight : (I → IsingSpin) → ℝ) | |
| (hweight : ∀ eta, 0 ≤ weight eta) | |
| (hlaw : ∀ eta, mu {x | ∀ i, x (p i) = eta i} = ENNReal.ofReal (weight eta)) | |
| (f : (I → IsingSpin) → ℝ) : | |
| ∫ x, f (fun i => x (p i)) ∂mu = ∑ eta : I → IsingSpin, weight eta * f eta := by | |
| refine integral_finiteLaw (X := fun x : TreeVertex d → IsingSpin => fun i => x (p i)) | |
| (Measurable.of_eval (fun index => measurable_pi_apply (p index))) weight hweight ?_ f | |
| intro eta | |
| convert hlaw eta using 2 | |
| ext x | |
| simp only [Set.mem_ofPred_eq, funext_iff] | |
| theorem integral_spinPath_of_cylinderLaw {d n : ℕ} {theta : ℝ} | |
| (hlo : -1 ≤ theta) (hhi : theta ≤ 1) | |
| {mu : Measure (TreeVertex d → IsingSpin)} [IsFiniteMeasure mu] | |
| (p : Fin (n+1) → TreeVertex d) | |
| (hlaw : ∀ eta : Fin (n+1) → IsingSpin, | |
| mu {x | ∀ i, x (p i) = eta i} = | |
| ENNReal.ofReal ((1/2 : ℝ) * spinPathWeight theta n eta)) : | |
| ∫ x, spinSign (x (p 0)) * spinSign (x (p (Fin.last n))) ∂mu = theta ^ n := by | |
| calc | |
| _ = ∑ eta : Fin (n+1) → IsingSpin, | |
| ((1/2 : ℝ) * spinPathWeight theta n eta) * | |
| (spinSign (eta 0) * spinSign (eta (Fin.last n))) := | |
| integral_finiteSpinCoordinates p _ | |
| (fun eta => mul_nonneg (by norm_num) (spinPathWeight_nonneg hlo hhi n eta)) | |
| hlaw _ | |
| _ = theta ^ n := by | |
| simpa only [mul_assoc] using spinPathWeight_correlation theta n | |
| end Problem367 | |
| end | |
| end OAI | |