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 of a real variable with respect to a real function g is denoted by

and defined to be the limit, as the norm of the partition

of the interval [a, b] approaches zero, of the approximating sum

where c_i is in the i-th subinterval [x_i, x_i+1]. The two functions f and g are respectively called the integrand and the integrator.

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 c_i in [x_i, x_i+1],

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 P_ε, meaning that P arises from P_ε 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 P_ε such that for every partition P that refines P_ε,

for every choice of points c_i in [x_i, x_i+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.

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