Theorems · Theorem · real analysis

intervalIntegral.hasFDerivAt_integral_of_dominated_of_fderiv_le

∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {μ : MeasureTheory.Measure ℝ} {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace ℝ E] [inst_3 : NormedSpace 𝕜 E] {H : Type u_3} [inst_4 : NormedAddCommGroup H]
  [inst_5 : NormedSpace 𝕜 H] {s : Set H} {a b : ℝ} {bound : ℝ → ℝ} {F : H → ℝ → E} {F' : H → ℝ → H →L[𝕜] E} {x₀ : H},
  s ∈ nhds x₀ →
    (∀ᶠ (x : H) in nhds x₀, MeasureTheory.AEStronglyMeasurable (F x) (μ.restrict (Set.uIoc a b))) →
      IntervalIntegrable (F x₀) μ a b →
        MeasureTheory.AEStronglyMeasurable (F' x₀) (μ.restrict (Set.uIoc a b)) →
          (∀ᵐ (t : ℝ) ∂μ, t ∈ Set.uIoc a b → ∀ x ∈ s, ‖F' x t‖ ≤ bound t) →
            IntervalIntegrable bound μ a b →
              (∀ᵐ (t : ℝ) ∂μ, t ∈ Set.uIoc a b → ∀ x ∈ s, HasFDerivAt (fun x => F x t) (F' x t) x) →
                HasFDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀

Differentiation under integral of x ↦ ∫ F x a at a given point x₀, assuming F x₀ is integrable, x ↦ F x a is differentiable on a neighborhood of x₀ for ae a with derivative norm uniformly bounded by an integrable function (the neighborhood independent of a), and F x is ae-measurable for x in a possibly smaller neighborhood of x₀.

Defined in

Mathlib.Analysis.Calculus.ParametricIntervalIntegral

Cited by

0 results in Mathlib

Foundations

Depth 264 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpaceNormedSpaceNormedAddCommGroupNormedSpace