# Riemann-Stieltjes integral

In

mathematics, the

**Riemann-Stieltjes integral** of a

real-valued function

*f* of a real variable with respect to a nondecreasing real function

*g* is denoted by

and defined to be the limit as the mesh of a partition

*P* of the interval [

*a*,

*b*] approaches zero, of the sum

where

*c*_{i} is in the

*i*th subinterval [

*x*_{i},

*x*_{i+1}]. In order that this Riemann-Stieltjes integral exist it is necessary that

*f* and

*g* do not share any points of discontinuity in common. The two functions

*f* and

*g* are respectively called the integrand and the integrator.

If *g* should happen to be everywhere differentiable, then the integral is no different from

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

### What if *g* is not monotone?

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

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

holds if

*g* is

*any* cumulative probability distribution function on the real line, no matter how ill-behaved.