measure_smul — Mathlib

English

A NNReal scalar scales the Stieltjes measure: (c • f).measure = c • f.measure.

Русский

Дробно-негационная скалярная константа масштабирует меру Стилетса: (c • f).measure = c • f.measure.

LaTeX

$$$(c \\\\in \\\\mathbb{R}_{≥0})\\\\(f) \\\\Rightarrow (c \\\\cdot f) \\\\mathrm{measure} = c \\\\cdot f \\\\mathrm{measure}$$$

Lean4

@[simp]
theorem measure_smul (c : ℝ≥0) (f : StieltjesFunction) : (c • f).measure = c • f.measure :=
  by
  refine Measure.ext_of_Ioc _ _ (fun a b _ ↦ ?_)
  simp only [measure_Ioc, Measure.smul_apply]
  change ofReal (c * f b - c * f a) = c • ofReal (f b - f a)
  rw [← _root_.mul_sub, ENNReal.ofReal_mul zero_le_coe, ofReal_coe_nnreal, ← smul_eq_mul]
  rfl

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