Table of contents |
2 What if g is not monotone? 3 Application to probability theory |
If g should happen to be everywhere differentiable, then the integral is no different from
Properties and relation to the Riemann integral
However, g may have jump discontinuities, or may have derivative zero almost everywhere while still being continuous and nonconstant (for example, g could be the celebrated Cantor function), in either of which cases the Riemann-Stieltjes integral is not captured by any expression involving derivatives of g.
The Riemann-Stieltjes integral admits integration by parts in the form
Somewhat more generally, one may define a Riemann-Stieltjes integral with respect to any function g of bounded variation, since every such function can be written uniquely as a difference between two nondecreasing functions; the integral is the corresponding difference between two Riemann-Stieltjes integrals with respect to nondecreasing functions.
If g is the cumulative probability distribution function of a random variable X that has a probability density function with respect to Lebesgue measure, and f is any function for which the expected value E(|f(X)|) is finite, then, as is well-known to students of probability theory, the probability density function of X is the derivative of g and we have
Application to probability theory
But this formula does not work if X does not have a probability density function with respect to Lebesgue measure. In particular, it does not work if the distribution of X is discrete (i.e., all of the probability is accounted for by point-masses), and even if the cumulative distribution function g is
continuous, it does not work if g fails to be absolutely continuous (again, the Cantor function may serve as an example of this failure). But the identity