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 of a real variable on the interval with respect to another real-to-real function is denoted by Its definition uses a sequence of partitions of the interval The integral, then, is defined to be the limit, as the mesh (the length of the longest subinterval) of the partitions approaches , of the approximating sum where is in the -th subinterval . 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 norm(P) < δ, and for every choice of points ci in [xi, xi+1], The Riemann–Stieltjes integral admits integration by parts in the form and the existence of either integral implies the existence of the other. On the other hand, a classical result shows that the integral is well-defined if f is α-Hölder continuous and g is β-Hölder continuous with α + β > 1 . If is bounded on , increases monotonically, and is Riemann integrable, then the Riemann–Stieltjes integral is related to the Riemann integral by For a step function where , if is continuous at , then 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 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.