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 _