Riemann–Stieltjes integral
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.[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.
Application to functional analysis[edit]
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.[4]
Existence of the integral[edit]
The best simple existence theorem states that if f is continuous and g is of bounded variation on [a, b], then the integral exists.[5][6][7]
Because of the integration by part formula, the integral exists also if the condition on f and g are inversed, that is, if f is of bounded variation and g is continuous.
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 there are other cases as well.
Examples and special cases[edit]
Differentiable g(x)[edit]
Given a which is continuously differentiable over it can be shown that there is the equality
