not_gradient_ae_congr_for_arbitrary_measure

Plain-English statement

- There is no unrestricted rule transporting classical gradients across almost-everywhere equality. The counterexample uses `δ₀`, the zero function, and the identity function: their values agree at the atom, while their gradients there are respectively zero and one.

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Lean statement

theorem not_gradient_ae_congr_for_arbitrary_measure :
    ¬ ∀ (μ : Measure ℝ) (f g : ℝ → ℝ),
      f =ᵐ[μ] g → ∀ᵐ x ∂μ, gradient f x = gradient g x := by
  intro h
  let f : ℝ → ℝ := fun _ => 0
  let g : ℝ → ℝ := fun x => x
  have hfg : f =ᵐ[Measure.dirac 0] g := by
    rw [ae_dirac_eq]
    rfl
  have hgrad := h (Measure.dirac 0) f g hfg
  have hgrad0 : gradient f 0 = gradient g 0 := by
    rw [ae_dirac_eq] at hgrad
    exact hgrad
  have hf0 : gradient f 0 = 0 := by
    simp [f]
  have hg0 : gradient g 0 = 1 := by
    exact (hasDerivAt_id (𝕜 := ℝ) 0).hasGradientAt.gradient
  rw [hf0, hg0] at hgrad0
  norm_num at hgrad0

end AutoSamplingTheory.TechnicalLemmas.Analysis.GradientAECongruence

Open AutoSamplingTheory/TechnicalLemmas/Analysis/GradientAECongruence.lean:23published source at ed709a7f320d

Proof architecture

prevent unsound transfer of a classical score from an explicit smooth log-ratio representative to an arbitrary a.e.-equal canonical llr representative

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Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.
Almost-everywhere hypotheses depend on the stated measure and representative.