Theorems · Theorem · probability

ProbabilityTheory.IsGaussian.map_rotation_eq_self_of_forall_strongDual_eq_zero

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : MeasurableSpace E] [BorelSpace E]
  {μ : MeasureTheory.Measure E} [ProbabilityTheory.IsGaussian μ] [SecondCountableTopology E] [CompleteSpace E],
  (∀ (L : StrongDual ℝ E), ∫ (x : E), L x ∂μ = 0) →
    ∀ (θ : ℝ), MeasureTheory.Measure.map (⇑(ContinuousLinearMap.rotation θ)) (μ.prod μ) = μ.prod μ

For a centered Gaussian measure μ, the product measure μ.prod μ is invariant under rotation. The hypothesis ∀ L : StrongDual ℝ E, μ[L] = 0 is equivalent to the simpler μ[id] = 0, but at this point we don't know yet that μ has a first moment so we can't use it. See map_rotation_eq_self.

Defined in

Mathlib.Probability.Distributions.Gaussian.Fernique

Cited by

2 results in Mathlib

Foundations

Depth 299 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceProbabilityTheory.IsGaussianSecondCountableTopologyCompleteSpace