toIsProbabilityMeasure — Mathlib

English

The gamma pdf is measurable.

Русский

Гамма-плотность является измеримой.

LaTeX

$$$\text{Measurable}(\gammaPDF(a,r))$$$

Lean4

/-- A Gaussian measure is a probability measure. -/
instance toIsProbabilityMeasure {E : Type*} [TopologicalSpace E] [AddCommMonoid E] [Module ℝ E] {mE : MeasurableSpace E}
    (μ : Measure E) [IsGaussian μ] : IsProbabilityMeasure μ where
  measure_univ :=
    by
    have : μ.map (0 : StrongDual ℝ E) Set.univ = 1 := by simp [IsGaussian.map_eq_gaussianReal]
    simpa [Measure.map_apply (by fun_prop : Measurable (0 : StrongDual ℝ E)) .univ] using this

Похожие утверждения