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

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

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

1 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

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 Cantor function or the question mark 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

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.

## See also

Last updated: 05-13-2005 07:56:04