Riemann–Stieltjes integral
[1] 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.
The integral, then, is defined to be the limit, as the mesh (the length of the longest subinterval) of the partitions approaches
is taken to be monotone (or at least of bounded variation) and right-semicontinuous (however this last is essentially convention).
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], The Riemann–Stieltjes integral admits integration by parts in the form
[2] On the other hand, a classical result[3] shows that the integral is well-defined if f is α-Hölder continuous and g is β-Hölder continuous with α + β > 1 .
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 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.
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.
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 there are other cases as well.
A 3D plot, with x, f(x), and g(x) all along orthogonal axes, leads to a geometric interpretation of the Riemann–Stieltjes integral.
[8] If the g(x)-x plane is horizontal and the f(x)-direction is pointing upward, then the surface to be considered is like a curved fence.
The fence is the section of the g(x)-sheet (i.e., the g(x) curve extended along the f(x) axis) that is bounded between the g(x)-x plane and the f(x)-sheet.
The Riemann-Stieltjes integral is the area of the projection of this fence onto the f(x)-g(x) plane — in effect, its "shadow".
The values of x for which g(x) has the steepest slope g'(x) correspond to regions of the fence with the greater projection and thereby carry the most weight in the integral.
the fence has a rectangular "gate" of width 1 and height equal to f(s).
Thus the gate, and its projection, have area equal to f(s), the value of the Riemann-Stieltjes integral.
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 of the interval [a,b].
This generalization plays a role in the study of semigroups, via the Laplace–Stieltjes transform.
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] .
can still be defined in cases where f and g have a point of discontinuity in common.
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
has jump discontinuities, or may have derivative zero almost everywhere while still being continuous and increasing (for example,
used in the study of neural networks, called a rectified linear unit (ReLU).
Cavalieri's principle can be used to calculate areas bounded by curves using Riemann–Stieltjes integrals.
