Let | denote the integral sign. Quoting from Priestly:
"...when we have two deterministic functions g(t),F(t), the Riemann-Stieltjes integral
R = |ab g(t)dF(t)
is defined as the limiting value of the discrete summation"
(sum from i=1 to i=n of) g(ti)[F(ti)-F(ti-1)]
for t1=a and tn=b as n goes to infinity and "as max(ti-ti-1)->0."
If F(t) is differentiable, then the above integral is the same as the regular integral R=|ab g(t)F'(t) dt, but the Reimann-Stieltjes integral can be defined in many cases even when F() is not differentiable.
One of the most common uses is when F() is a cdf.
Examples: The expectation of a random variable can be written:
mu=| xf(x) dx
if f(x) is the pdf. It can also be written:
mu=| x dF(x)
where F(x) is the cdf. The two are equivalent for a continuous distribution, but notice that for a discrete one (e.g. a coin flip, with X=0 for heads and X=1 for tails) the second, Riemann-Stieltjes, formulation is well defined but no pdf exists to calculate the first one.
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