File size: 1,234 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 | import OAI.Probability.InvariantIsing.Arrays.ReplicaProductContinuity
namespace OAI
/-! A bounded continuous conditional residual is zero when all bounded
continuous total-overlap tests annihilate it. -/
noncomputable section
open MeasureTheory IsingPerceptron
namespace InvariantIsing
theorem ae_zero_of_continuous_overlap_tests {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (p : Ω → ℝ) (hp : Measurable p)
(f : ℝ → ℝ) (hf : Continuous f) {B : ℝ} (hB : 0 ≤ B) (hb : ∀ r, |f r| ≤ B)
(htest : ∀ Φ : ℝ → ℝ, Continuous Φ → ∀ C : ℝ, 0 ≤ C → (∀ r, |Φ r| ≤ C) →
(∫ x, Φ (p x) * f (p x) ∂μ) = 0) : ∀ᵐ x ∂μ, f (p x) = 0 := by
have hi : Integrable (fun x => f (p x) ^ 2) μ := by
apply integrable_of_measurable_abs_le ((hf.measurable.comp hp).pow_const 2)
intro x
rw [abs_pow]
exact pow_le_pow_left₀ (abs_nonneg _) (hb (p x)) 2
have hz : (∫ x, f (p x) ^ 2 ∂μ) = 0 := by
simpa only [pow_two] using htest f hf B hB hb
have he := (integral_eq_zero_iff_of_nonneg (fun x => sq_nonneg (f (p x))) hi).mp hz
filter_upwards [he] with x hx
exact sq_eq_zero_iff.mp hx
end InvariantIsing
end
end OAI
|