measurable_deriv_with_param — Mathlib

English

For a fixed f with parametric dependence and a measure μ on α × 𝕜, the derivative deriv( f p.1 ) p.2 is AEMeasurable with respect to μ.

Русский

Для заданной зависимости через параметр и меры μ на α × 𝕜 производная deriv( f p.1 ) p.2 является AEMеас measurable.

LaTeX

$$$$\\AEMeasurable (\\lambda p: (\\alpha\\times 𝕜), \\ deriv (f p.1) p.2) \\; μ$$$$

Lean4

theorem measurable_deriv_with_param [LocallyCompactSpace 𝕜] [MeasurableSpace 𝕜] [OpensMeasurableSpace 𝕜]
    [MeasurableSpace F] [BorelSpace F] {f : α → 𝕜 → F} (hf : Continuous f.uncurry) :
    Measurable (fun (p : α × 𝕜) ↦ deriv (f p.1) p.2) := by
  simpa only [fderiv_deriv] using measurable_fderiv_apply_const_with_param 𝕜 hf 1

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