∀ {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 : NormedSpace ℝ E]
  {ν : MeasureTheory.Measure E} [ProbabilityTheory.IsGaussian μ] [ProbabilityTheory.IsGaussian ν],
  ∫ (x : E), id x ∂μ = ∫ (x : E), id x ∂ν →
    ProbabilityTheory.covarianceBilinDual μ = ProbabilityTheory.covarianceBilinDual ν → μ = ν