Riemann-Stieltjes explanation, usage, and application?

Question

Does anyone here know a good explanatory text for the Riemann-Stieltjes integral? I sort of get how there is a funcion alpha which we are using in place of the x-axis, and partitioning it instead, and somewhere it said it was like a density, but I don't understand it in full. Also, any text on applications to help differentiate it from the standard Riemann integral would be helpful.

? · Accepted Answer

Suppose you have one of those lottery tickets that come up in prob / stats class.

For some set of outcomes you get a reward, and for others you don't

the expected value of the ticket then it the probability of each event * the pay off for each event.

if f(x) is your payoff function and p(x) is your probability density function, then

E[f(x)] = ∫[-infinity to infinity] f(x) p(x) dx

if p(x) is continuous, then the integral is the same as the ordinary integrals you have always done.

but if you have discontinuities, you need something that is a little bit more flexible.

_______

if p(x) is your probability density formula.
then your integral is

E[f(x)] = ∫[-infinity to infinity] f(x) p(x) dx

but if you have a cumulative distribution function ... call it P(x)

E[f(x)] = ∫[-infinity to infinity] f(x) d p(x)

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Riemann-Stieltjes explanation, usage, and application?

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