Theorems · Theorem · abstract harmonic analysis

MeasureTheory.integral_convolution

∀ {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 : RCLike 𝕜]
  [inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜 E'] [inst_6 : NormedSpace ℝ F] [inst_7 : NormedSpace 𝕜 F]
  [inst_8 : MeasurableSpace G] {μ ν : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [CompleteSpace F]
  [inst_10 : AddGroup G] [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] [μ.IsAddRightInvariant] [MeasurableAdd₂ G]
  [MeasurableNeg G] [inst_16 : NormedSpace ℝ E] [inst_17 : NormedSpace ℝ E'] [CompleteSpace E] [CompleteSpace E'],
  MeasureTheory.Integrable f ν →
    MeasureTheory.Integrable g μ →
      ∫ (x : G), MeasureTheory.convolution f g L ν x ∂μ = (L (∫ (x : G), f x ∂ν)) (∫ (x : G), g x ∂μ)

Defined in

Mathlib.Analysis.Convolution

Cited by

1 results in Mathlib

Foundations

Depth 267 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupRCLikeNormedSpaceNormedSpaceNormedSpaceNormedSpaceMeasurableSpaceCompleteSpaceAddGroupMeasureTheory.SFiniteMeasureTheory.SFiniteMeasureTheory.Measure.IsAddRightInvariantMeasurableAdd₂MeasurableNegNormedSpaceNormedSpaceCompleteSpaceCompleteSpace