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a2ee19a | 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 | import Mathlib
import OAI.Probability.Perceptron.Brownian.CountableGaussianField
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal BigOperators
namespace SphericalPerceptronFreeEnergy
def resampleGaussianCoordinate (i : ℕ) (p : ℝ×(ℕ → ℝ)) : ℕ → ℝ :=
Function.update p.2 i p.1
lemma resampleGaussianCoordinate_measurable (i : ℕ) :
Measurable (resampleGaussianCoordinate i) := by
unfold resampleGaussianCoordinate
fun_prop
lemma resampleGaussianCoordinate_measurePreserving (i : ℕ) :
MeasurePreserving (resampleGaussianCoordinate i)
((gaussianReal 0 1).prod countableGaussianLaw) countableGaussianLaw := by
refine ⟨resampleGaussianCoordinate_measurable i,?_⟩
apply Measure.eq_infinitePi
intro s t ht
rw [Measure.map_apply (resampleGaussianCoordinate_measurable i)
(MeasurableSet.pi s.countable_toSet (fun j _ => ht j))]
classical
by_cases hi : i ∈ s
· have he : (resampleGaussianCoordinate i) ⁻¹' Set.pi s t =
t i ×ˢ Set.pi (s.erase i) t := by
ext p
simp only [mem_preimage,mem_prod,Set.mem_pi,Finset.mem_coe,resampleGaussianCoordinate]
constructor
· intro hp
refine ⟨?_,fun j hj => ?_⟩
· simpa only [Function.update_self] using hp i hi
· simpa only [Function.update_of_ne (Finset.mem_erase.mp hj).1] using
hp j (Finset.mem_of_mem_erase hj)
· rintro ⟨hp,hq⟩ j hj
by_cases he : j=i
· subst j
simpa only [Function.update_self] using hp
· simpa only [Function.update_of_ne he] using hq j (Finset.mem_erase.mpr ⟨he,hj⟩)
rw [he,Measure.prod_prod,countableGaussianLaw,Measure.infinitePi_pi _ (fun j _ => ht j)]
exact Finset.mul_prod_erase s (fun j => gaussianReal 0 1 (t j)) hi
· have he : (resampleGaussianCoordinate i) ⁻¹' Set.pi s t =
Set.univ ×ˢ Set.pi s t := by
ext p
simp only [mem_preimage,mem_prod,true_and,mem_univ,Set.mem_pi,Finset.mem_coe,resampleGaussianCoordinate]
constructor <;> intro hp j hj
· simpa only [Function.update_of_ne (ne_of_mem_of_not_mem hj hi)] using hp j hj
· simpa only [Function.update_of_ne (ne_of_mem_of_not_mem hj hi)] using hp j hj
rw [he,Measure.prod_prod,measure_univ,one_mul,countableGaussianLaw,
Measure.infinitePi_pi _ (fun j _ => ht j)]
end SphericalPerceptronFreeEnergy
end
end OAI
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