measurable_stieltjesFunction — Mathlib

English

Same as above: the limit at +∞ is 1 for the Stieltjes function.

Русский

То же самое: предел при +∞ равен 1 для функции Стилетджесса.

LaTeX

$$$$ \\lim_{x \\to +\\infty} (hf.stieltjesFunction a) x = 1 $$$$

Lean4

theorem measurable_stieltjesFunction (x : ℝ) : Measurable fun a ↦ hf.stieltjesFunction a x :=
  by
  have : (fun a ↦ hf.stieltjesFunction a x) = fun a ↦ ⨅ r : { r' : ℚ // x < r' }, f a ↑r :=
    by
    ext1 a
    rw [← StieltjesFunction.iInf_rat_gt_eq]
    congr with q
    rw [stieltjesFunction_eq]
  rw [this]
  exact .iInf (fun q ↦ hf.measurable.eval)

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