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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.

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Probability is an area that uses it often. – toypajme Dec 21 '12 at 23:33
Skip the chapter. – Qiaochu Yuan Dec 21 '12 at 23:39
There is a point to Riemann-Stieltjes integration, but the point is fairly far away from complex line integrals. As Qiaochu says, it is very likely that this material can be safely skipped. What is the text? – Pete L. Clark Dec 22 '12 at 1:46
It can be convenient in analytic number theory. – Eric Naslund Dec 22 '12 at 4:39
@Eric: I'd be interested to hear about that, could you please expand that into an answer? – user5501 Dec 22 '12 at 4:54

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.
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Can't all of those examples be handled by the Lebesgue integral? (At least the first and third one.) – mrf Dec 22 '12 at 8:43
I see it the complete other way around: The Lebesgue-Stieltjes integral is a special case of the Lebesgue integral. And as the Wikipedia page notes: "The Lebesgue-Stieltjes integral $\int_a^b f(x) \,dg(x)$ is defined as the Lebesgue integral of $f$ with respect to the measure $μ_g$ in the usual way." – kahen Dec 22 '12 at 12:55
@kahen I see your point, I was thinking more of as the integral with respect to Lebesgue measure is a special case of the Lebesgue-Stieltjes integral. – Learner Dec 22 '12 at 13:00
In any case, being a special case of the Lebesgue integral does not diminish the attractiveness of the Lebesgue-Stieltjes integral, both conceptually and practically. – Learner Dec 22 '12 at 13:40

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