File size: 3,063 Bytes
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 | 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
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