This short note gives an introduction to the Riemann-Stieltjes integral on R and Rn. Some natural and important applications in probability theory are discussed. The reason for discussing the Riemann-Stieltjes integral instead of the more general Lebesgue and LebesgueStieltjes integrals are that most applications in elementary probability theory are satisfactorily covered by the Riemann-Stieltj...

Using the notion of the truncated variation we obtain a new theorem on the existence and estimation of the Riemann-Stieltjes integral. As a special case of this theorem we obtain an improved version of the Loéve-Young inequality for the Riemann-Stieltjes integrals driven by irregular signals. Using this result we strengthen some results of Lyons on the existence of solutions of integral equatio...

Several inequalities of Grüss type for the Stieltjes integral with various type of integrand and integrator are introduced. Some improvements inequalities are proved. Applications to the approximation problem of the Riemann–Stieltjes integral are also pointed out.

By Theorem 2.1 the product φ1φ2 is of bounded variation and by Theorem 2.24 all three Riemann–Stieltjes integrals involved exist as the limit of Riemann–Stieltjes sums as |Γ| → 0. In particular we can find a δ1 > 0 such that ∣∣∣∣∫ b a fφ2 dφ1 −RΓ(fφ2, φ1) ∣∣∣∣ < ε , ∣∣∣∣∫ b a fφ1 dφ2 −RΓ(fφ1, φ2) ∣∣∣∣ < ε for all partitions with |Γ| < δ1. Let Γ = {xi}i=0 be any partition of [a, b] with intermed...

in this paper, a generalization of trapezoid inequality for functions of two independent variables with bounded variation and some applications are given.

We present a variety of series representations of the Stieltjes and related constants, the Stieltjes constants being the coefficients of the Laurent expansion of the Hurwitz zeta function ζ(s, a) about s = 1. Additionally we obtain series and integral representations of a sum Sγ(n) formed as an alternating binomial series from the Stieltjes constants. The slowly varying sum Sγ(n) + n is an impo...

This paper presents a formal proof of the Riesz representation theorem in the PVS theorem prover. The Riemann Stieltjes integral was defined in PVS, and the theorem relies on this integral. In order to prove the Riesz representation theorem, it was necessary to prove that continuous functions on a closed interval are Riemann Stieltjes integrable with respect to any function of bounded variation...