Theorems · Theorem · real analysis

intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_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] {a b : ℝ} {bound : ℝ → ℝ} {F F' : 𝕜 → 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' 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, HasDerivAt (fun x => F x t) (F' x t) x) →
                IntervalIntegrable (F' x₀) μ a b ∧
                  HasDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ 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 differentiable on a neighborhood of x₀ for ae a (with a neighborhood independent of a) with derivative uniformly bounded by an integrable function, and F x is ae-measurable for x in a possibly smaller neighborhood of x₀.

Defined in

Mathlib.Analysis.Calculus.ParametricIntervalIntegral

Cited by

1 results in Mathlib

Foundations

Depth 265 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpaceNormedSpace