Modified Riemann sums of Riemann-Stieltjes integrable functions

TL;DR

This paper introduces modified Riemann sums for Riemann-Stieltjes integrable functions, demonstrates their convergence, and provides explicit computations under different assumptions, motivated by physical considerations.

Contribution

It presents a new approach to Riemann sums for Riemann-Stieltjes integrals, extending classical methods with modifications inspired by physical applications.

Findings

Modified Riemann sums converge for Riemann-Stieltjes integrable functions.

Explicit formulas for the sums are derived under various assumptions.

The approach is motivated by physical considerations.

Abstract

In this paper, motivated by physical considerations, we introduce the notion of modified Riemann sums of Riemann-Stieltjes integrable functions, show that they converge, and compute them explicitely under various assumptions.