Theorems · Theorem · measure theory

LocallyIntegrable.ae_hasDerivAt_integral

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

The (global) interval version of the Lebesgue Differentiation Theorem: if f : ℝ → E is locally integrable, then for almost every x, for any c : ℝ, 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 262 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace