Remarks on the existence of the Riemann-Stieltjes integral

The Riemann-stieltjes Integral on Time Scales

We study the process of integration on time scales in the sense of Riemann-Stieltjes. Analogues of the classical properties are proved for a generic time scale, and examples are given.

On Approximation of the Riemann–stieltjes Integral and Applications

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.

A Note on the Convergence of Random Riemann and Riemann-Stieltjes Sums to the Integral

Convergence in probability of random Riemann sums of a Lebesgue integrable function on [0, 1) to the integral has been proved. In this article we generalize the result to an abstract probability space under some natural conditions and we show L1convergence rather than convergence in probability..

A new theorem on the existence of the Riemann-Stieltjes integral and an improved version of the Loéve-Young inequality

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...

A Companion of Ostrowski’s Inequality for the Riemann-stieltjes Integral