Theorems · Theorem · abstract harmonic analysis

BddAbove.continuous_convolution_right_of_integrable

∀ {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 : NormedSpace ℝ F]
  [inst_9 : AddGroup G] [inst_10 : TopologicalSpace G] [IsTopologicalAddGroup G] [BorelSpace G]
  [FirstCountableTopology G] [SecondCountableTopologyEither G E'],
  BddAbove (Set.range fun x => ‖g x‖) →
    MeasureTheory.Integrable f μ → Continuous g → Continuous (MeasureTheory.convolution f g L μ)

The convolution is continuous if one function is integrable and the other is bounded and continuous.

Defined in

Mathlib.Analysis.Convolution

Cited by

2 results in Mathlib

Foundations

Depth 253 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupNontriviallyNormedFieldNormedSpaceNormedSpaceNormedSpaceMeasurableSpaceNormedSpaceAddGroupTopologicalSpaceIsTopologicalAddGroupBorelSpaceFirstCountableTopologySecondCountableTopologyEither