Theorems · Theorem · real analysis

intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_right

• #15 of the 100 theorems: Fundamental Theorem of Integral Calculus
• 1000+ list: Fundamental theorem of calculus

∀ {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {c : E}
  {a b : ℝ},
  IntervalIntegrable f MeasureTheory.volume a b →
    StronglyMeasurableAtFilter f (nhds b) MeasureTheory.volume →
      Filter.Tendsto f (nhds b ⊓ MeasureTheory.ae MeasureTheory.volume) (nhds c) →
        HasStrictDerivAt (fun u => ∫ (x : ℝ) in a..u, f x) c b

Fundamental theorem of calculus-1, strict differentiability in the right endpoint. If f : ℝ → E is integrable on a..b and f x has a finite limit c almost surely at b, then u ↦ ∫ x in a..u, f x has derivative c at b in the sense of strict differentiability.

Defined in

Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Cited by

3 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace