Riemann-Stieltjes integral

In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes.

Definition
The Riemann-Stieltjes integral of a real-valued function f of a real variable with respect to a real function g is denoted by


 * $$\int_a^b f(x) \, dg(x)$$

and defined to be the limit, as the mesh of the partition P of the interval [a, b] approaches zero, of the approximating sum


 * $$\sum_{x_i\in P}f(c_i)(g(x_{i+1})-g(x_i))$$

where ci is in the i-th subinterval [xi, xi+1]. The two functions f and g are respectively called the integrand and the integrator. Most commonly, g will be nondecreasing, but this is not required. In order that this Riemann-Stieltjes integral exist it is necessary that f and g do not share any points of discontinuity.

An alternative, and slightly more general, definition of the Riemann-Stieltjes integral uses the same approximating sums as above, but takes the limit to be a Moore-Smith limit on the directed set of partitions of [a, b]. That is, take the limit as more and more division points are inserted into the partition. With this definition, an integral can exist when f and g share points of discontinuity, as long as they are not discontinuous from the same side at the same point.

For another formulation of integration that is much more general, see Lebesgue integration. It is notable however, that if improper Riemann-Stieltjes integrals are allowed, the Lebesgue integral is not strictly more general.

Properties and relation to the Riemann integral
If g should happen to be everywhere differentiable, then the integral may still be different from the Riemann integral


 * $$\int_a^b f(x) g'(x) \, dx,$$

for example, if the derivative is unbounded. But if the derivative is continuous, they will be the same. This condition is also satisfied if g is the (Lebesgue) integral of its derivative; in this case g is said to be absolutely continuous.

However, g may have jump discontinuities, or may have derivative zero almost everywhere while still being continuous and increasing (for example, g could be the Cantor function or Minkowski's 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


 * $$\int_a^b f(x) \, dg(x)=f(b)g(b)-f(a)g(a)-\int_a^b g(x) \, df(x).$$

and the existence of the integral on the left implies the existence of the integral on the right.

Existence of the integral
The best simple existence theorem states that if f is continuous and g is of bounded variation on [a, b], then the integral exists. Note that g is of bounded variation if and only if it is the difference between two monotone functions. If g is not of bounded variation, then there will be continuous functions which cannot be integrated with respect to g.

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


 * $$E(f(X))=\int_{-\infty}^\infty f(x)g'(x)\, dx.$$

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


 * $$E(f(X))=\int_{-\infty}^\infty f(x)\, dg(x)$$

holds if g is any cumulative probability distribution function on the real line, no matter how ill-behaved.

Application to functional analysis
The Riemann-Stieltjes integral appears in the original formulation of F. Riesz theorem which represents the dual space of the Banach space $$ C[a,b]$$ of continuous functions in an interval $$[a,b]$$ as Riemann-Stieltjes integrals against functions of bounded variation  (later, that theorem was reformulated in terms of measures).

Also, the Riemann-Stieltjes integral appears in the formulation of the spectral theorem for (non-compact) self-adjoint (or more generally, normal) operators in a Hilbert space (in this theorem, the integral is considered with respect to a so-called spectral family of projections).

[see the book by F. Riesz for details]