# Riemann-Stieltjes integral

In mathematics, the**Riemann-Stieltjes integral**is a generalization of the Riemann integral. 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

*a*,

*b*] approaches zero, of the sum

*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.

For another formulation of integration that is more general, see Lebesgue integration.

Table of contents |

2 What if g is not monotone?3 Application to probability theory 4 See also |

### Properties and relation to the Riemann integral

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

*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.

### Application to probability theory

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

*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

*g*is

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