Theorems · Theorem · probability

ProbabilityTheory.IsGaussian.map_rotation_eq_self

∀ {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] [CompleteSpace E] [SecondCountableTopology E],
  ∫ (x : E), id 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.

Defined in

Mathlib.Probability.Distributions.Gaussian.Fernique

Cited by

0 results in Mathlib

Foundations

Depth 312 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceProbabilityTheory.IsGaussianCompleteSpaceSecondCountableTopology