Theorems · Theorem · real analysis

intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_lip

∀ {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] {a b : ℝ} {bound : ℝ → ℝ} {F : 𝕜 → ℝ → E} {F' : ℝ → E} {x₀ : 𝕜}
  {s : Set 𝕜},
  s ∈ nhds x₀ →
    (∀ᶠ (x : 𝕜) in nhds x₀, MeasureTheory.AEStronglyMeasurable (F x) (μ.restrict (Set.uIoc a b))) →
      IntervalIntegrable (F x₀) μ a b →
        MeasureTheory.AEStronglyMeasurable F' (μ.restrict (Set.uIoc a b)) →
          (∀ᵐ (t : ℝ) ∂μ, t ∈ Set.uIoc a b → LipschitzOnWith (Real.nnabs (bound t)) (fun x => F x t) s) →
            IntervalIntegrable bound μ a b →
              (∀ᵐ (t : ℝ) ∂μ, t ∈ Set.uIoc a b → HasDerivAt (fun x => F x t) (F' t) x₀) →
                IntervalIntegrable F' μ a b ∧
                  HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀

Derivative under integral of x ↦ ∫ F x a at a given point x₀ : 𝕜, 𝕜 = ℝ or 𝕜 = ℂ, assuming F x₀ is integrable, x ↦ F x a is locally Lipschitz on a neighborhood of x₀ for ae a (with a neighborhood independent of a) with integrable Lipschitz bound, 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

RCLikeNormedAddCommGroupNormedSpaceNormedSpace