isGaussian_gaussianReal — Mathlib

English

The gamma pdf is strongly measurable (dup).

Русский

Гамма-плотность сильно измерима (дубликат).

LaTeX

$$$\text{StronglyMeasurable}(\gammaPDFReal(a,r))$$$

Lean4

/-- A real Gaussian measure is Gaussian. -/
instance isGaussian_gaussianReal (m : ℝ) (v : ℝ≥0) : IsGaussian (gaussianReal m v) where
  map_eq_gaussianReal
    L := by
    rw [gaussianReal_map_continuousLinearMap]
    simp only [integral_continuousLinearMap_gaussianReal, variance_continuousLinearMap_gaussianReal, Real.coe_toNNReal']
    congr
    rw [Real.toNNReal_mul (by positivity), Real.toNNReal_coe]
    congr
    simp only [left_eq_sup]
    positivity

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