memLp_dual — Mathlib

English

For all a > 0, r > 0, x ∈ ℝ, gammaPDFReal(a, r, x) ≥ 0.

Русский

Для всех a > 0, r > 0 и x ∈ ℝ, gammaPDFReal(a, r, x) ≥ 0.

LaTeX

$$$0 \le \gammaPDFReal(a,r,x)$$$

Lean4

theorem memLp_dual (μ : Measure E) [IsGaussian μ] (L : StrongDual ℝ E) (p : ℝ≥0∞) (hp : p ≠ ∞) : MemLp L p μ :=
  by
  suffices MemLp (id ∘ L) p μ from this
  rw [← memLp_map_measure_iff (by fun_prop) (by fun_prop), IsGaussian.map_eq_gaussianReal L]
  convert memLp_id_gaussianReal p.toNNReal
  simp [hp]

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