How to prove that $\int_a^b = \int_a^c + \int_c^b$ for Riemann-Stieltjes integrals

Question

Let $f$ be Riemann-Stieltjes integrable with respect to $g$ on $[a,b]$. By Riemann-Stieltjes integrable I mean that $\forall\varepsilon>0\exists\delta$ such that when the norm of any partition $P$ of $[a,b]$ is smaller than $\delta$ then $\mid S(P,T,f,g)-\int_{a}^{b}f(x)dg(x)\mid<\varepsilon$ where $T$ is an arbitrary choice of tags. I'm trying to show that for $\forall c\in[a,b]$ $\int_{a}^{b}f(x)dg(x) = \int_{a}^{c}f(x)dg(x) + \int_{c}^{b}f(x)dg(x)$. This seems to be a bit tricky to show as opposed to some other linear properties. The problem comes down to the fact that the partition of choice is arbitrary as long as its norm is sufficiently small. So if $c$ is not an endpoint of a subinterval of the partition then the Riemann-Stieltjes sum cannot be split into a sum over $[a,c]$ and $[c,b]$ and that way be treated easily. Assuming I found a way to do this, then I would still have to prove that the two integrals on the right hand side exist. To my understanding the proof of this is quite a bit easier if we require $g$ to be monotone or perhaps a bound variation which is an entirely new thing for me. However, I don't want to consider special cases. Perhaps there are cases where the integrals exists without g being monotone or a bound variation, I would like to include those as well in the proof. I'm thinking that I perhaps have to find an equivalent definition that is easier to work with and prove it that way but I don't really know how to go about doing so. Some help would be greatly appreciated.

Answer 1

Expanding my comment into an answer.

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You need to use Cauchy condition for integrability:

Cauchy's Condition: $f$ is integrable with respect to $g$ on $[a, b] $ if and only if for every $\epsilon >0$ there is a $\delta>0$ such that $|S(P_1,T_1,f,g)-S(P_2,T_2,f,g)|<\epsilon $ for all partitions $P_1,P_2$ of $[a, b] $ with norm less than $\delta $ and $T_1,T_2$ being arbitrary tags for $P_1,P_2$ respectively.

Using this we can prove the integrability of $f$ wrt $g$ on $[c, d] \subseteq [a, b] $ if $f$ is integrable wrt $g$ on $[a, b] $. Let's begin with an arbitrary $\epsilon >0$ and choose $\delta>0$ which meets the Cauchy condition for $f, g$ on $[a, b]$. If $P_1,P_2$ are arbitrary partitions of $[c, d] $ with norm less than $\delta$ and $T_1,T_2$ corresponding choice of tags then it is possible to extend these to partitions $P'_1,P'_2$ of $[a, b] $ and choose tags $T'_1,T'_2$ accordingly such that $$S(P_1,T_1,f,g)-S(P_2,T_2,f,g)=S(P'_1,T'_1,f,g)-S(P'_2,T'_2,f,g)$$ and then for Cauchy condition the same $\delta$ works for $f, g$ on $[c, d]$.

Now using Riemann Stieltjes sums you can prove your desired result.