Definition

The Riemann–Stieltjes integral of a real-valued function f of a real variable with respect to a real function g is denoted by

   $$ \int_a^b f(x) \, dg(x)$$

and defined to be the limit, as the mesh of the partition

   $$ P=\{ a = x_0 < x_1 < \cdots < x_n = b\}$$

of the interval $[a, b]$ approaches zero, of the approximating sum

    $$S(P,f,g) = \sum_{i=0}^{n-1} f(c_i)(g(x_{i+1})-g(x_i))$$

where $c_i$ is in the $i$-th subinterval $[x_i, x_{i+1}]$. The two functions $f$ and $g$ are respectively called the {\bf integrand} and the {\bf integrator}.

The {\bf limit} is a number $A$, the value of the Riemann–Stieltjes integral. The meaning of the limit: Given $\varepsilon > 0$, then there exists $\delta > 0$ such that for every partition $ P=\{ a = x_0 < x_1 < \cdots < x_n = b\}$ with $mesh(P) < \delta$, and for every choice of points $c_i$ in $[x_i, x_{i+1}]$,

    $$|S(P,f,g)-A| < \varepsilon. \, $$


