Theorems · Theorem · global analysis

ContDiffBump.dist_normed_convolution_le

∀ {G : Type uG} {E' : Type uE'} [inst : NormedAddCommGroup E'] {g : G → 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'] {φ : ContDiffBump 0} [BorelSpace G] [inst_7 : FiniteDimensional ℝ G] [μ.IsAddHaarMeasure] {x₀ : G}
  {ε : ℝ},
  MeasureTheory.AEStronglyMeasurable g μ →
    (∀ x ∈ Metric.ball x₀ φ.rOut, dist (g x) (g x₀) ≤ ε) →
      dist (MeasureTheory.convolution (φ.normed μ) g (ContinuousLinearMap.lsmul ℝ ℝ) μ x₀) (g x₀) ≤ ε

If φ is a normed bump function, approximate (φ ⋆ g) x₀ if g is near g x₀ on a ball with radius φ.rOut around 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