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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 35 36 37 38 | import OAI.Probability.InvariantIsing.Haar.PublishedInputs
import Mathlib.Probability.Distributions.Gaussian.Fernique
namespace OAI
/-! Square integrability of the actual Lipschitz observables in the Gaussian input. -/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace InvariantIsing
lemma gaussian_lipschitz_memLp_two {d : ℕ} {L : ℝ≥0}
{f : EuclideanSpace ℝ (Fin d) → ℝ} (hf : LipschitzWith L f) :
MemLp f 2 (stdGaussian (EuclideanSpace ℝ (Fin d))) := by
have hi : MemLp (fun x : EuclideanSpace ℝ (Fin d) => x) 2
(stdGaussian (EuclideanSpace ℝ (Fin d))) := IsGaussian.memLp_id _ 2 (by norm_num)
have hc : LipschitzWith L (fun x => f x-f 0) := by
apply LipschitzWith.of_dist_le_mul
intro x y
simpa only [dist_eq_norm, sub_sub_sub_cancel_right] using hf.dist_le_mul x y
have hh := (hc.comp_memLp (by simp) hi).add
(memLp_const (f 0) : MemLp (fun _ : EuclideanSpace ℝ (Fin d) => f 0) 2
(stdGaussian (EuclideanSpace ℝ (Fin d))))
convert hh using 1
funext x
simp
lemma gaussian_lipschitz_integrable {d : ℕ} {L : ℝ≥0}
{f : EuclideanSpace ℝ (Fin d) → ℝ} (hf : LipschitzWith L f) :
Integrable f (stdGaussian (EuclideanSpace ℝ (Fin d))) :=
(gaussian_lipschitz_memLp_two hf).integrable (by norm_num)
end InvariantIsing
end
end OAI
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