instModuleNNReal — Mathlib

English

If f is a monotone function, then the constructed StieltjesFunction from hf is a module over R.

Русский

Если f монотонна, то построенная функция Стильтьеса образует модуль над ℝ.

LaTeX

$$$$ \\text{Monotone } f \\Rightarrow \\; \\text{Monotone } (\\text{stieltjesFunction } f). $$$$

Lean4

instance : Module ℝ≥0 StieltjesFunction
    where
  smul c
    f :=
    { toFun := fun x ↦ c * f x
      mono' := f.mono.const_mul c.2
      right_continuous' := fun x ↦ (f.right_continuous x).const_smul c.1 }
  one_smul _ := ext fun _ ↦ one_mul _
  mul_smul _ _ _ := ext fun _ ↦ mul_assoc _ _ _
  smul_zero _ := ext fun _ ↦ mul_zero _
  smul_add _ _ _ := ext fun _ ↦ mul_add _ _ _
  add_smul _ _ _ := ext fun _ ↦ add_mul _ _ _
  zero_smul _ := ext fun _ ↦ zero_mul _

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