Theorems · Theorem · real analysis

intervalIntegral.derivWithin_integral_of_tendsto_ae_left

∀ {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 →
    ∀ {s t : Set ℝ} [intervalIntegral.FTCFilter a (nhdsWithin a s) (nhdsWithin a t)],
      StronglyMeasurableAtFilter f (nhdsWithin a t) MeasureTheory.volume →
        Filter.Tendsto f (nhdsWithin a t ⊓ MeasureTheory.ae MeasureTheory.volume) (nhds c) →
          autoParam (UniqueDiffWithinAt ℝ s a) intervalIntegral.derivWithin_integral_of_tendsto_ae_left._auto_1 →
            derivWithin (fun u => ∫ (x : ℝ) in u..b, f x) s a = -c

Fundamental theorem of calculus: if f : ℝ → E is integrable on a..b and f x has a finite limit c almost surely as x tends to a from the right or from the left, then the right (resp., left) derivative of u ↦ ∫ x in u..b, f x at a equals -c.

Defined in

Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Cited by

0 results in Mathlib

Foundations

Depth 267 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpaceintervalIntegral.FTCFilter