Theorems · Theorem · global analysis

ContDiffBump.convolution_tendsto_right

∀ {G : Type uG} {E' : Type uE'} [inst : NormedAddCommGroup E'] [inst_1 : MeasurableSpace G]
  {μ : MeasureTheory.Measure G} [inst_2 : NormedSpace ℝ E'] [inst_3 : NormedAddCommGroup G] [inst_4 : NormedSpace ℝ G]
  [CompleteSpace E'] [BorelSpace G] [inst_7 : FiniteDimensional ℝ G] [μ.IsAddHaarMeasure] {ι : Type u_1}
  {φ : ι → ContDiffBump 0} {g : ι → G → E'} {k : ι → G} {x₀ : G} {z₀ : E'} {l : Filter ι},
  Filter.Tendsto (fun i => (φ i).rOut) l (nhds 0) →
    (∀ᶠ (i : ι) in l, MeasureTheory.AEStronglyMeasurable (g i) μ) →
      Filter.Tendsto (Function.uncurry g) (l ×ˢ nhds x₀) (nhds z₀) →
        Filter.Tendsto k l (nhds x₀) →
          Filter.Tendsto
            (fun i => MeasureTheory.convolution ((φ i).normed μ) (g i) (ContinuousLinearMap.lsmul ℝ ℝ) μ (k i)) l
            (nhds z₀)

(φ i ⋆ g i) (k i) tends to z₀ as i tends to some filter l if * φ is a sequence of normed bump functions such that (φ i).rOut tends to 0 as i tends to l; * g i is μ-a.e. strongly measurable as i tends to l; * g i x tends to z₀ as (i, x) tends to l ×ˢ 𝓝 x₀; * k i tends to x₀.

Defined in

Mathlib.Analysis.Calculus.BumpFunction.Convolution

Cited by

1 results in Mathlib

Foundations

Depth 283 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupMeasurableSpaceNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasure