Theorems · Theorem · abstract harmonic analysis

MeasureTheory.ConvolutionExistsAt.of_norm

∀ {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [inst : NormedAddCommGroup E]
  [inst_1 : NormedAddCommGroup E'] [inst_2 : NormedAddCommGroup F] {f : G → E} {g : G → E'}
  [inst_3 : NontriviallyNormedField 𝕜] [inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜 E'] [inst_6 : NormedSpace 𝕜 F]
  (L : E →L[𝕜] E' →L[𝕜] F) [inst_7 : MeasurableSpace G] {μ : MeasureTheory.Measure G} [inst_8 : AddGroup G]
  [MeasurableAdd₂ G] [MeasurableNeg G] [MeasureTheory.SFinite μ] [μ.IsAddRightInvariant] {x₀ : G},
  MeasureTheory.ConvolutionExistsAt (fun x => ‖f x‖) (fun x => ‖g x‖) x₀ (ContinuousLinearMap.mul ℝ ℝ) μ →
    MeasureTheory.AEStronglyMeasurable f μ →
      MeasureTheory.AEStronglyMeasurable g μ → MeasureTheory.ConvolutionExistsAt f g x₀ L μ

If ‖f‖ *[μ] ‖g‖ exists, then f *[L, μ] g exists.

Defined in

Mathlib.Analysis.Convolution

Cited by

1 results in Mathlib

Foundations

Depth 230 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupNontriviallyNormedFieldNormedSpaceNormedSpaceNormedSpaceMeasurableSpaceAddGroupMeasurableAdd₂MeasurableNegMeasureTheory.SFiniteMeasureTheory.Measure.IsAddRightInvariant