Theorems · Theorem · measure theory

MeasureTheory.charFun_conv

∀ {E : Type u_3} [inst : MeasurableSpace E] {μ ν : MeasureTheory.Measure E} [inst_1 : NormedAddCommGroup E]
  [inst_2 : InnerProductSpace ℝ E] [BorelSpace E] [SecondCountableTopology E] [MeasureTheory.IsFiniteMeasure μ]
  [MeasureTheory.IsFiniteMeasure ν] (t : E),
  MeasureTheory.charFun (μ.conv ν) t = MeasureTheory.charFun μ t * MeasureTheory.charFun ν t

The characteristic function of a convolution of measures is the product of the respective characteristic functions.

Defined in: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic
Cited by: 2 results in Mathlib
Foundations: Depth 266 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: MeasurableSpace NormedAddCommGroup InnerProductSpace BorelSpace SecondCountableTopology MeasureTheory.IsFiniteMeasure MeasureTheory.IsFiniteMeasure

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Cites37

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Realstatement and proof · cited by 25,697
NormedAddCommGroupstatement and proof · cited by 15,752
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
Complexstatement and proof · cited by 5,565
InnerProductSpacestatement and proof · cited by 3,523
Filter.Eventuallyproof · cited by 3,134
MeasureTheory.aeproof · cited by 2,352
MeasureTheory.integralproof · cited by 1,779
Complex.ofRealproof · cited by 1,654
BorelSpacestatement and proof · cited by 1,602
Inner.innerproof · cited by 1,089

Cited by2

Results whose statement or proof uses this declaration.

ProbabilityTheory.gaussianReal_conv_gaussianRealproof · cited by 1
ProbabilityTheory.IndepFun.charFun_map_add_eq_mulproof · cited by 1