Theorems · Theorem · real analysis

intervalIntegral.integral_hasFDerivWithinAt

∀ {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E}
  {la lb : Filter ℝ} {a b : ℝ},
  IntervalIntegrable f MeasureTheory.volume a b →
    StronglyMeasurableAtFilter f la MeasureTheory.volume →
      StronglyMeasurableAtFilter f lb MeasureTheory.volume →
        ∀ {s t : Set ℝ} [intervalIntegral.FTCFilter a (nhdsWithin a s) la]
          [intervalIntegral.FTCFilter b (nhdsWithin b t) lb],
          Filter.Tendsto f la (nhds (f a)) →
            Filter.Tendsto f lb (nhds (f b)) →
              HasFDerivWithinAt (fun p => ∫ (x : ℝ) in p.1..p.2, f x)
                ((ContinuousLinearMap.snd ℝ ℝ ℝ).smulRight (f b) - (ContinuousLinearMap.fst ℝ ℝ ℝ).smulRight (f a))
                (s ×ˢ t) (a, b)

Let f be a measurable function integrable on a..b. The function (u, v) ↦ ∫ x in u..v, f x has derivative (u, v) ↦ v • f b - u • f a within s × t at (a, b), where s ∈ {Iic a, {a}, Ici a, univ} and t ∈ {Iic b, {b}, Ici b, univ} provided that f tends to f a and f b at the filters la and lb from the following table. In most cases this assumption is definitionally equal ContinuousAt f _ or ContinuousWithinAt f _ _. | s | la | t | lb | | - | - | | - | | Iic a | 𝓝[≤] a | Iic b | 𝓝[≤] b | | Ici a | 𝓝[>] a | Ici b | 𝓝[>] b | | {a} | ⊥ | {b} | ⊥ | | univ | 𝓝 a | univ | 𝓝 b |

Defined in

Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Cited by

0 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceCompleteSpaceintervalIntegral.FTCFilterintervalIntegral.FTCFilter