Theorems · Theorem · probability

ProbabilityTheory.IsGaussian.eq_dirac_of_variance_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] [CompleteSpace E] [SecondCountableTopology E],
  (∀ (L : StrongDual ℝ E), ProbabilityTheory.variance (⇑L) μ = 0) → μ = MeasureTheory.Measure.dirac (∫ (x : E), x ∂μ)

A Gaussian measure with variance zero is a Dirac.

Defined in

Mathlib.Probability.Distributions.Gaussian.Fernique

Cited by

1 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceProbabilityTheory.IsGaussianCompleteSpaceSecondCountableTopology