| 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 | |