Riemann Stieltjes integral without monotonic increasing $\alpha$ I am recently Reading Rudin's Principles of Mathematical Analysis. I have a converse question to this post.
I feel like the monotonical increase of $\alpha$ is not very carefully dealt with in the chapter. For example, Theorem 6.22 (integration by parts, $\int fdg=fg - \int gdf$) has no assumption on the monotonicity of $f$ or $g$, but its proof relies on 6.21, 6.7(c), and eventually the definition 6.2 of $L$ and $U$, where we assume $\alpha$ is monotonically increasing. 
I know for any $\alpha$ of bounded variation, there exist monotonically increasing $\alpha_1$ and $\alpha_2$ such that $\alpha=\alpha_1-\alpha_2$, and $\int fd\alpha$ can be decomposed as $\int fd\alpha_1-\int fd \alpha_2$, and most of the results still hold. But I feel like Rudin's approach to this Riemann-Stieltjes part should be somehow re-done. Or is there an easy way to disentangle the requirement of monotonicity of $\alpha$ without using BV?
 A: Baby Rudin's treatment of Riemann-Stieltjes integral is not upto the mark. It tries to simplify things unnecessarily. Rudin's books have somehow become more popular but one should also have a look at other options while studying analysis. 
Apostol's Mathematical Analysis handles it in a more thorough manner. The topic of integration by parts is handled in Theorem 7.6, page 144 and it says that (wording changed to add more detail instead of using Apostol's notation) :

Theorem: Let the functions $f:[a, b] \to\mathbb {R}, g:[a, b] \to\mathbb {R} $ be bounded on $[a, b]$. Also let $f$ be Riemann-Stieltjes integrable with respect to $g$ on $[a, b] $ ie $\int_{a} ^{b} f(x) \, dg(x) $ exists. Then $g$ is also Riemann-Stieltjes integrable with respect to $f$ on $[a, b] $ ie $\int_{a} ^{b} g(x)\, df(x) $ also exists and we have $$\int_{a}^{b} f(x) \, dg(x) +\int_{a} ^{b} g(x) \, df(x) =f(b) g(b) - f(a) g(a) $$

There is no mention of any other conditions on $f, g$ apart from them being bounded. The proof of the above theorem is not complicated at all and is based on just the definitions.
Apostol defines the integral only for bounded functions. Much of the theory is developed for integrators which are of bounded variation, but if a result holds for other more general integrators then it is mentioned explicitly at the end of the theorem (and sometimes a proof is also provided for the general case).
The requirement of monotone integrators is primarily due to the definition chosen by Rudin. He uses upper and lower Darboux sums and then you must have monotone integrators which can be generalized further to deal with integrators of bounded variation.
Apostol prefers to use Riemann-Stieltjes sums which don't need monotone integrators and hence are more general.
