Q:

What is the point of Riemann-Stieltjes integration?

My book on complex analysis dedicates an entire chapter to this integral in order to motivate/define complex line integrals. Having spent a semester on integration theory, I am not to keen on learning about yet another integral when I am perfectly satisfied with the Lebesgue integral. My understanding is that the Riemann-Stieltjes integral is just a generalization of the Riemann integral. I simply don't see why I need it for this course or why I would ever need it at all. Some insight would be greatly appreciated.

A:

I can give three reasons:

It is extremely useful in probability theory. When, for example, you would like to write the expectation of a random variable, sometimes, people spend time explaining the differences between discrete and continuous variables and give different formulas. Sometimes, they completely ignore random variables that are neither discrete nor continuous in order not to give even more formulas. With the Riemann-Stieltjes integrals, it all boils down to one formula. This makes so many other things in probability theory clearer and more lucid.
The Ito or stochastic integral is really a generalization of the Riemann-Stieltjes integral. It is useful for solving stochastic differential equations, partial differential equations and many other things.
It makes many arguments in physics where calculus with respect to Dirac functions is done more rigorous.

