Theorems · Theorem · measure theory

IntervalIntegrable.ae_hasDerivAt_integral

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {a b : ℝ},
  IntervalIntegrable f MeasureTheory.volume a b →
    ∀ᵐ (x : ℝ), x ∈ Set.uIcc a b → ∀ c ∈ Set.uIcc a b, HasDerivAt (fun x => ∫ (t : ℝ) in c..x, f t) (f x) x

The (local) interval version of the Lebesgue Differentiation Theorem: if f : ℝ → E is interval integrable on a..b, then for almost every x ∈ uIcc a b, for any c ∈ uIcc a b, the derivative of ∫ (t : ℝ) in c..x, f t at x is equal to f x.

Defined in

Mathlib.MeasureTheory.Integral.IntervalIntegral.LebesgueDifferentiationThm

Cited by

1 results in Mathlib

Foundations

Depth 263 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace