Theorems · Theorem · probability

ProbabilityTheory.IsGaussian.ext_iff

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [SecondCountableTopology E] [CompleteSpace E]
  [inst_3 : MeasurableSpace E] [inst_4 : BorelSpace E] {μ : MeasureTheory.Measure E} [inst_5 : InnerProductSpace ℝ E]
  {ν : MeasureTheory.Measure E} [ProbabilityTheory.IsGaussian μ] [ProbabilityTheory.IsGaussian ν],
  μ = ν ↔
    ∫ (x : E), id x ∂μ = ∫ (x : E), id x ∂ν ∧ ProbabilityTheory.covarianceBilin μ = ProbabilityTheory.covarianceBilin ν

Two Gaussian measures are equal if and only if they have same mean and same covariance. This is IsGaussian.ext_iff_covarianceBilinDual specialized to Hilbert spaces.

Defined in

Mathlib.Probability.Distributions.Gaussian.CharFun

Cited by

0 results in Mathlib

Foundations

Depth 314 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupSecondCountableTopologyCompleteSpaceMeasurableSpaceBorelSpaceInnerProductSpaceProbabilityTheory.IsGaussianProbabilityTheory.IsGaussian