Rieman-Stieltjes integration "by parts" Let us define a partition $a=x_0<x_1<...<x_n=b$ of interval $[a,b]$ and let us define the Riemann-Stieltjes integral $\int_a^b fdg$ of a function $f:[a,b]\to\mathbb{C}$, or $f:[a,b]\to\mathbb{R}$, as the limit, as the maximum of the leght of the partition intervals approaches 0, which must be independent from the partition chosen, of the Riemann-Stieltjes sum$$\sum_{i=1}^n f(\xi_i)[g(x_i)-g(x_{i-1})]$$where $\xi_i\in[x_{i-1},x_i)$ and $g:[a,b]\to\mathbb{R}$ is a function of bounded variation and continuous from the left$^1$. This is Kolmogorov and Fomin's definition in Элементы теории функций и функционального анализа (p. 362 here). I know -thanks to a very kind user of this forum- the following equality, where if one of the two integrals exists does the other one does too:$$\int_a^b fdg=f(b)g(b)-f(a)g(a)-\int_a^bgdf$$
I have read a proof of the equality in Hille and Phillips' Functional analysis and semi-groups, but it uses two partitions such that $t_0=s_0=a$, $t_{n+1}=s_n=b$ and $t_i≤s_i≤t_{i+1}$ and the identity $\sum^n_{i=1}f(t_i)[g(s_i)−g(s_{i-1})]=f(t_{n+1})g(s_n)−f(t_0)g(s_0)−\sum^n_{i=0}g(s_i)[f(t_{i+1})−f(t_i)]$. I am not sure how to adapt a proof of this kind because


*

*$t_0,s_0,t_{n+1},s_n$ are set, while in Kolmogorov-Fomin's definition the limit of the Riemann-Stieltjes sum must be independent from the partition chosen;

*the arguments of the functions are allowed to be the right endpoints of the partition intervals.


My clumsy trial to modify Hille and Phillips' proof is here. How could the equality be proved to satisfy Kolmogorov and Fomin's definition of Riemann-Stieltjes integral? I do not even know whether Kolmogorov and Fomin's and Hille and Phillips' are equivalent for a real function of bounded variation. I thank anybody for any answer!!!
$^1$ Really, Kolmogorov and Fomin use $g:[a,b)\to\mathbb{R}$ continuous from the left and then set $g(b):=g(b^-)$, imposing that $g$ must be of bounded variation on $[a,b]$. Notwithstanding my efforts to understand why Kolmogorov and Fomin's definition of $g$ is not "$g:[a,b]\to\mathbb{R}$ of bounded variation and continuous from the left", I am convinced that theirs is equivalent to this last much more straightforward definition, which I therefore use.
 A: I offered a first "solution" by mistakenly assuming $f$ was continuous from the left instead of $g$. I don't see how to show that the integrals are equivalent for a general $f$ and bounded variation $g$ normalized to be continuous on the left. The English translation of the 1968 second edition contains the standard definition of Riemann-Stieljes integral with evaluation points $t_{k}^{\star}\in[t_{k-1},t{k}]$, instead of one with evaluation points $t_{k}^{\star}\in [t_{k-1},t_{k})$. So I would suggest using the standard definition instead. The old definition for $\int_a^b f\,dg$ requires $g$ to be of bounded variation and normalized to be left continuous, and those assumptions are not needed for the general integral; and such assumptions prevent you from easily obtaining the general integration-by-parts theorem which would require both $f$ and $g$ to be of bounded variation in order to use their old definition.
I strongly suspect there must be a difference, or the Authors would not have changed the definition to the standard one. Of course, another compelling reason to change the definition is that general Riemann-Stieltjes integrals don't need to be restricted to the case where one of the functions is of bounded variation; so general normalization of $g$ in $\int_{a}^{b}f\,dg$ cannot be assumed without assuming left-hand limits exist everywhere. Such limits exist for functions of bounded variation, but--as I mentioned--that assumption is not needed to define the integral.
Their original definition appears to have been motivated out of trying to connect the Lebesgue-Stieltjes integral with the Riemann-Stieltjes integrals. These two integrals are just not compatible. The Lebesgue-Stieltjes integral $\int_{a}^{b}f\,dg$ requires a measure $g$ coming from a function of bounded variation, while Riemann-Stieltjes does not require $g$ to be of bounded variation. Functions $f$ with discontinuities at an atom of $g$ can be integrated using Lesbesgue-Stieltjes, but this is not the case for the Riemann-Stieltjes $\int_{a}^{b}f\,dg$. These are significant differences, and there are others. So trying to closely wed the Riemann-Stieltjes definition to match Lebesgue-Stieltjes is not a good idea; even though it may help lead to Lebesgue-Stieltjes, the important generality of Riemann-Stieltjes is lost is the process. Any differences found when restricting to functions $g$ as you describe are likely to be related to the differences between Lebesgue-Stieltjes and Riemann-Stieltjes.
