Theorems · Theorem · probability

ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq_fun

∀ {Ω : Type u_1} {E : Type u_2} {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} [inst : NormedAddCommGroup E]
  [inst_1 : MeasurableSpace E] [BorelSpace E] {X : Ω → E} [inst_3 : NormedSpace ℝ E] (L : StrongDual ℝ E),
  ProbabilityTheory.HasGaussianLaw X P →
    MeasureTheory.charFunDual (MeasureTheory.Measure.map X P) L =
      Complex.exp (↑(∫ (ω : Ω), L (X ω) ∂P) * Complex.I - ↑(ProbabilityTheory.variance (fun ω => L (X ω)) P) / 2)

Defined in: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 297 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup MeasurableSpace BorelSpace NormedSpace

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

DFunLike.coestatement and proof · cited by 62,936
Realstatement and proof · cited by 25,697
RingHom.idstatement · cited by 18,349
NormedAddCommGroupstatement and proof · cited by 15,752
MeasurableSpacestatement and proof · cited by 13,106
NormedSpacestatement and proof · cited by 12,499
MeasureTheory.Measurestatement and proof · cited by 10,939
Complexstatement and proof · cited by 5,565
MeasureTheory.integralstatement and proof · cited by 1,779
Complex.ofRealstatement and proof · cited by 1,654
BorelSpacestatement and proof · cited by 1,602
Complex.Istatement and proof · cited by 866

Cited by1

Results whose statement or proof uses this declaration.

ProbabilityTheory.HasGaussianLaw.iIndepFun_of_covariance_strongDualproof · cited by 3