Overview
When the function {% F(x) %} with which a Riemann Stieltjes integral is taken with respect to is smooth, the Riemann Stieltjes integral can be shown to be equivalent to the following Riemann Integral.
{% \int g(x) dF(x) = \int g(x) \frac{dF(x)}{dx} dx %}
The standard Riemann integral equivalent to the Riemann-Stieltjes integral
{% \int g(x) dx = \int g(x) dF(x) %}
where {% F(x) = x %}.
Discontinuous Function
When the integrating function is not continuous, the Riemann Stieltjes integral can still be viewed as equivalent to a Riemann integral with the help of the Dirac Delta Function.
That is, if the derivative of the integrating function at the points of discontinuity are taken to be the Dirac Delta Function multiplied by the size of the discontinuity, then the integral is equivalent to its Riemann version.