Theorems · Theorem · abstract harmonic analysis

BddAbove.convolutionExistsAt

∀ {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 : AddCommGroup G]
  [MeasurableNeg G] [μ.IsAddLeftInvariant] [MeasurableAdd₂ G] [MeasureTheory.SFinite μ] {x₀ : G} {s : Set G},
  BddAbove ((fun i => ‖g i‖) '' (fun t => x₀ - t) ⁻¹' s) →
    MeasurableSet s →
      (Function.support fun t => (L (f t)) (g (x₀ - t))) ⊆ s →
        MeasureTheory.IntegrableOn f s μ →
          MeasureTheory.AEStronglyMeasurable g μ → MeasureTheory.ConvolutionExistsAt f g x₀ L μ

A sufficient condition to prove that f ⋆[L, μ] g exists. We assume that the integrand has compact support and g is bounded on this support (note that both properties hold if g is continuous with compact support). We also require that f is integrable on the support of the integrand, and that both functions are strongly measurable. This is a variant of BddAbove.convolutionExistsAt' in an abelian group with a left-invariant measure. This allows us to state the boundedness and measurability of g in a more natural way.

Defined in

Mathlib.Analysis.Convolution

Cited by

1 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupNontriviallyNormedFieldNormedSpaceNormedSpaceNormedSpaceMeasurableSpaceAddCommGroupMeasurableNegMeasureTheory.Measure.IsAddLeftInvariantMeasurableAdd₂MeasureTheory.SFinite