Theorems · Theorem · real analysis

intervalIntegral.fderivWithin_integral_of_tendsto_ae

∀ {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {ca cb : 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 ⊓ MeasureTheory.ae MeasureTheory.volume) (nhds ca) →
            Filter.Tendsto f (lb ⊓ MeasureTheory.ae MeasureTheory.volume) (nhds cb) →
              autoParam (UniqueDiffWithinAt ℝ s a) intervalIntegral.fderivWithin_integral_of_tendsto_ae._auto_1 →
                autoParam (UniqueDiffWithinAt ℝ t b) intervalIntegral.fderivWithin_integral_of_tendsto_ae._auto_3 →
                  fderivWithin ℝ (fun p => ∫ (x : ℝ) in p.1..p.2, f x) (s ×ˢ t) (a, b) =
                    (ContinuousLinearMap.snd ℝ ℝ ℝ).smulRight cb - (ContinuousLinearMap.fst ℝ ℝ ℝ).smulRight ca

Let f be a measurable function integrable on a..b. Choose s ∈ {Iic a, Ici a, univ} and t ∈ {Iic b, Ici b, univ}. Suppose that f tends to ca and cb almost surely at the filters la and lb from the table below. Then fderivWithin ℝ (fun p ↦ ∫ x in p.1..p.2, f x) (s ×ˢ t) is equal to (u, v) ↦ u • cb - v • ca. | 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