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. 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 Riemann–Stieltjes integral of a real-valued function f {displaystyle f} of a real variable with respect to a real function g {displaystyle g} is denoted by and defined to be the limit, as the norm (or mesh) of the partition (i.e. the length of the longest subinterval) of the interval approaches zero, of the approximating sum where c i {displaystyle c_{i}} is in the i-th subinterval . The two functions f {displaystyle f} and g {displaystyle g} are respectively called the integrand and the integrator. Typically g {displaystyle g} 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 g {displaystyle g} 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 , The Riemann–Stieltjes integral admits integration by parts in the form and the existence of either integral implies the existence of the other.