What does it mean to ''construct a Riemann-Stieltjes integral''? This question refers to an exercise (1.11c) from Reed & Simon's book on mathematical physics. In the first parts of this problem, the reader is introduced to the notion of functions of bounded variation:

Definition: A function $\alpha$ on $[0,1]$ is said to be of bounded variation if there is a $C$ such that $$\sum_{i=1}^{n-1} |\alpha(x_{i+1})-\alpha(x_i)|\leq C $$ for any sequence $0\leq x_1\leq \dots \leq x_n \leq 1 $.

We then start by defining $I_\alpha$ a function on the space of step functions on $[0,1]$, called $S[0,1]$. It acts on step functions (of the form $\sum_{i=1}^n s_i \chi_i$, where $s_i$ is a constant and $\chi_i$ is the characteristic function of the interval $[x_i,x_{i+1})$) as follows:
$$I_\alpha\left(\sum_{i=1}^n s_i \chi_i \right)=\sum_{i=1}^n s_i[\alpha(x_i)-\alpha(x_{i-1})]$$
I then was asked to proof that $I_\alpha$ is a bounded linear operator iff $\alpha$ is of bounded variation (fairly simple, so I managed to do so). The next part of the question then reads:

Let $\alpha$ be of bounded variation on $[0,1]$. Construct a Riemann-Stieltjes integral $\int f\ \mathrm{d}\alpha$

This leaves me mystified: What does it mean to 'construct an integral'? Looking through the material I already covered I found a relevant passage about the Riemann integral. This passage (on page 11) follows after it is shown that a function $I$ acting on $S[0,1]$, defined in a fashion completely analogous to $I_\alpha$ which I defined earlier (simply taking $\alpha(x)=x$), is a B.L.T. on $S[0,1]$.
The authors simply state that, since the real numbers are complete, this B.L.T. can be uniquely extended to the completion of $S$ (using the B.L.T. theorem, which was introduced earlier). "The extended transformation $\tilde{I}(f)$, restricted to $PC[0,1]$, is then called the Riemann integral." It is then remarked that the method of first defining a function on a dense set in the space of interest and then using the completion to extend it is more generally useful, and that this should be tried by the reader in problem 1.11 (not very enlightening...). 
However, it is left completely unspecified what exactly does one have to do in order to be able to say that one has 'constructed an integral'. I know that the B.L.T. theorem can be used to extend the function to the space of interest (I assume $PC$ once again?), but is there anything that concretely needs to be done other than 'waving the magic wand' and saying that the extension of my $I_\alpha$ acting on $S[0,1]$ is the Riemann-Stieltjes integral?
 A: If $\alpha$ is a function of bounded variation on $[a,b]$, and if $f$ is a continuous function, then the classical definition of the Riemann-Stieljes integral of $f$ with respect to $\alpha$ is
$$
             \int_{a}^{b}f(t)\,d\alpha(t) = \lim_{\|\mathscr{P}\|\rightarrow 0}
            \sum_{n=1}^{k}f(t_{k}^{\star})(\alpha(t_{k})-\alpha(t_{k-1}),
$$
where $\mathscr{P}$ is a partition $a = t_{0} < t_{1} < \cdots < t_{n}=b$ and $t_{k}^{\star} \in [t_{k-1},t_{k}]$ is the usual Riemann type partition for $[a,b]$. Such an integral limit exists for any continuous function $f$ if $\alpha$ is a function of bounded variation on $[a,b]$. The integral satisfies
$$
           \left|\int_{a}^{b}f(t)\,d\alpha(t)\right| \le \|f\|_{\infty}\|\alpha\|,
$$
where $\|f\|_{\infty}=\max_{a\le t\le b} |f(t)|$ and where $\|\alpha\|$ is the total variation for $\alpha$ on $[a,b]$. In other words,
$$
                I_{\alpha}(f) = \int_{a}^{b}f(t)\,d\alpha(t)
$$
is a bounded linear functional on $C[a,b]$. Or, using the Reed-Simon acronym, $I_{\alpha}$ is a B.L.T..
The exercise that this came from in Reed-Simon is where they ask you to define $I_{\alpha}$ on the linear space of step functions $S[a,b]$, to show $I_{\alpha}$ is bounded with respect to the max norm on $S[a,b]$ iff $\alpha$ is of bounded variation, and to use an argument to extend $I_{\alpha}$ uniquely by continuity to $C[a,b]$, knowing that $S[a,b]$ is dense in $C[a,b]$. That's what they mean by construct the integral. The continuous extension of $I_{\alpha}$ to $C[a,b]$ is their construction of the integral. This problem parallels the construction used earlier in the Chapter for the Riemann integral.
Note: This is not completely classical because the classical definition of the Riemann-Stieltjes integral $\int_{a}^{b}f(t)d\alpha(t)$ may not exist for a function $f$ which is not continuous at some $c \in (a,b)$, if $\alpha$ is also not continuous at $c$.
