I posted this question about a month ago: If $f$ is $g$-Riemann-Stieltjes integrable on $[a,b]$, prove that it's $g$-RS-integrable on $[a,c] \subset [a,b]$, and RRL explained to me that there are two non-equivalent definitions of the Riemann-Stieltjes integral, and provided a proof of what I needed with respect to one of those definitions. However, I'm looking for a proof using the following definition (and I was told to ask another question highlighting the fact that I want this definition to be used):

Let $f \in BV[a,b]$ and let $g:[a,b] \to \mathbb{C}$ be two complex functions, where $f$ is of bounded variation, and $g$ is arbitrary. Let $P:a=x_{0}<x_{1}<...<x_{n}=b$ be a partition of $[a,b]$ and $\xi = (\xi_{1},...,\xi_{n})$ such that $\xi_{k} \in [x_{k-1},x_{k}], k=\overline{1,n}$ - the tagged points of this partition. We define the Riemann-Stieltjes sum as $$\sigma(P, \xi, f, g) = \sum_{k=1}^{n} g(\xi_{k})[f(x_{k}) - f(x_{k-1})].$$

Also, define $\Delta(P) = \max_{1 \leq k \leq n} (x_{k}-x_{k-1})$. We say that $I \in \mathbb{C}$ is the Riemann-Stieltjes integral of $g$ with respect to $f$ over $[a,b]$ if: for any $\varepsilon > 0$, there exists a $\delta > 0$ such that, for all partitions $P$ of $[a,b]$ with $\Delta(P) < \delta$ and any set of tagged points of $P$, we have $$|\sigma(P, \xi, f, g) - I| < \varepsilon.$$

We write $$I = \int_{a}^{b} g\hspace{1mm}df.$$

In What is the definition of Riemann-Stieltjes integrability?, it is shown that this definition is not equivalent to the definition using upper and lower RS-integrals.

With respect to this definition, I am looking for a proof of the following claim:

If $f \in BV[a,b]$ and $g:[a,b] \to \mathbb{C}$ is Riemann-Stieltjes integrable on $[a,b]$ with respect to $f$, and if $a < c < b$, prove that $g$ is Riemann-Stieltjes integrable on $[a,c]$ and $[c, b]$ with respect to $f$ and that $$\int_{a}^{b} g\hspace{1mm}df = \int_{a}^{c} g\hspace{1mm}df + \int_{c}^{b} g\hspace{1mm}df.$$

I have offered my thoughts on this question in the linked question, however, I haven't made any progress since. I would be grateful if someone were to provide a clear proof, or a hint to me.


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