I am studying the Riemann Stieltjes on Tom Apostol's book mathematical analysis second edition and I have a the following question.
Given $[a,b]$ we define a partition of this interval to be a set $P = \{a = x_0, ..., x_n = b\}$
Suppose $ f,g:[a,b]\rightarrow \mathbb{R}$ and $f$ is Riemman integrable with respect to $g$ on $[a,b]$ ie for every $\varepsilon > 0$ there exists $P_{\varepsilon}$ partition of $[a,b]$ such that for every $P\subseteq P_{\varepsilon}$ and any choice of points $\{t_k\}$, (the points within each subinterval of the partition), we have that:
$\left | S(P,f,g) - A \right | < \varepsilon$ for some real $A$ which is what we denote by $\int_{a}^{b} fdg(x)$.
Now, my question is, 1) if given the fact that $f$ is integrable with respect to $g$ on $[a,b]$ can we say that $f$ is integrable with respect to $g$ on $(a,b)$ or $(a,b]$ or $[a,b)$? 2) If yes is it direct from the definition??(please be rigorous and not intuitive) 3) Are the integrals always equal?
Is it direct from the definition that the integrals on those intervals are equal? What i see is that the Riemann sums are not equal since
$S(P,f,g) = \sum_{k=0}^{n} f(t_k)(g(x_k)-g(x_{k-1})) $
so when the endpoints are not included we don't have the $g(a)/g(b)$ terms.
In terms of Lebesgue integration one could say that if $g$ is nonnegative and $f$ is measurable then $f1_{[a,b]} = f1_{[a,b)}$ a.s with respect to the measure induced by g, so the corresponding integrals are equal.
Here in Riemann Stieltjes integration does the definition imply directly a same argument or we need some Lebesgue-related proposition??
Thank you for your patience!