In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the Lebesgue integral, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability.
of the interval [a, b] approaches zero, of the approximating sum
where is in the i-th subinterval [xi, xi+1]. The two functions and are respectively called the integrand and the integrator. Typically is taken to be monotone (or at least of bounded variation) and right-semicontinuous (however this last is essentially convention). We specifically do not require to be continuous, which allows for integrals that have point mass terms.
The "limit" is here understood to be a number A (the value of the Riemann–Stieltjes integral) such that for every ε > 0, there exists δ > 0 such that for every partition P with mesh(P) < δ, and for every choice of points ci in [xi, xi+1],
Generalized Riemann–Stieltjes integral
A slight generalization, introduced by Pollard (1920) and now standard in analysis, is to consider in the above definition partitions P that refine another partition Pε, meaning that P arises from Pε by the addition of points, rather than from partitions with a finer mesh. Specifically, the generalized Riemann–Stieltjes integral of f with respect to g is a number A such that for every ε > 0 there exists a partition Pε such that for every partition P that refines Pε,
for every choice of points ci in [xi, xi+1].
This generalization exhibits the Riemann–Stieltjes integral as the Moore–Smith limit on the directed set of partitions of [a, b] (McShane 1952). Hildebrandt (1938) calls it the Pollard–Moore–Stieltjes integral.
The Riemann–Stieltjes integral can be efficiently handled using an appropriate generalization of Darboux sums. For a partition P and a nondecreasing function g on [a, b] define the upper Darboux sum of f with respect to g by
and the lower sum by
Then the generalized Riemann–Stieltjes of f with respect to g exists if and only if, for every ε > 0, there exists a partition P such that
Furthermore, f is Riemann–Stieltjes integrable with respect to g (in the classical sense) if
See Graves (1946, Chap. XII, §3).
Of course, the standard Riemann integral is an example of the Riemann-Stieltjes integral if is taken as .
Given a which is continuously differentiable over it can be shown that there is the equality
where the integral on the right-hand side is the standard Riemann-integral. This of course assumes that is already integrable for this Riemann-Stieltjes integral. More generally, the Riemann integral equals the Riemann–Stieltjes integrals if is the (Lebesgue) integral of its derivative; in this case is said to be absolutely continuous. It may be the case that has jump discontinuities, or may have derivative zero almost everywhere while still being continuous and increasing (for example, could be the Cantor function or Devil's staircase), in either of which cases the Riemann–Stieltjes integral is not captured by any expression involving derivatives of g.
Consider the function called the RELU. Then the Riemann-Stieltjes can be evaluated as
where the integral on the right-hand side is the standard Riemann integral.
The Riemann–Stieltjes integral admits integration by parts in the form
and the existence of either integral implies the existence of the other (Hille & Phillips 1974, §3.3).
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. A function g is of bounded variation if and only if it is the difference between two (bounded) monotone functions. If g is not of bounded variation, then there will be continuous functions which cannot be integrated with respect to g. In general, the integral is not well-defined if f and g share any points of discontinuity, but this sufficient condition is not necessary.
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 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. In particular, no matter how ill-behaved the cumulative distribution function g of a random variable X, if the moment E(Xn) exists, then it is equal to
Application to functional analysis
The Riemann–Stieltjes integral appears in the original formulation of F. Riesz's 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.
The Riemann–Stieltjes integral also 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 spectral family of projections. See Riesz & Sz. Nagy (1955) for details.
An important generalization is the Lebesgue–Stieltjes integral which generalizes the Riemann–Stieltjes integral in a way analogous to how the Lebesgue integral generalizes the Riemann integral. If improper Riemann–Stieltjes integrals are allowed, the Lebesgue integral is not strictly more general than the Riemann–Stieltjes integral.
The Riemann–Stieltjes integral also generalizes to the case when either the integrand ƒ or the integrator g take values in a Banach space. If g : [a,b] → X takes values in the Banach space X, then it is natural to assume that it is of strongly bounded variation, meaning that
the supremum being taken over all finite partitions
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