In Rudin's Principle of Mathematical Analysis chapter 6, Rudin discusses change of variables with Riemann-Stieltjes Integration.
Can this theorem be manipulated and applied to 'improper' integral that Rudin mentions in Exercise 7 and 8?
What if a function Φ such that for any B>0, Φ maps [0,B] onto [0,c] such that C(B):=inf{Φ(x):0≤x≤B} is monotonically decreasing and C(B) converges to 0 as B goes to infinity, Φ is differentiable on any [0,B], and Φ' is Riemann-Integrable?
Does such function exist? If so, is it easily applicable?
Sure. For example,
$$\Phi(x) = \frac{1}{x}-1$$
maps $[0,1]$ to $[0,\infty)$.