[Paper Review] Existence and computation of Riemann-Stieltjes integrals through Riemann integrals

Rodrigo López Pouso|arXiv (Cornell University)|Jul 11, 2011

Advanced Thermodynamics and Statistical Mechanics3 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition for the existence and computation of Riemann–Stieltjes integrals of bounded functions via Riemann integrals. It proves that if the integrator $ G $ is an indefinite Riemann integral of a Riemann integrable function $ g $, then $ f $ is Riemann–Stieltjes integrable with respect to $ G $ if and only if the product $ fg $ is Riemann integrable, in which case the Riemann–Stieltjes integral equals the Riemann integral of $ fg $.

ABSTRACT

We study the existence of Riemann-Stieltjes integrals of bounded functions against a given integrator. We are also concerned with the possibility of computing the resulting integrals by means of related Riemann integrals. In particular, we present a new generalization to the well-known formula for continuous integrands and continuously differentiable integrators.

Motivation & Objective

• To establish a necessary and sufficient condition for the existence of Riemann–Stieltjes integrals of bounded functions with respect to a given integrator $ G $.
• To extend the classical formula $ \int f\,dG = \int fG'\,dx $ beyond continuously differentiable $ G $, allowing for more general integrators.
• To provide a method for computing Riemann–Stieltjes integrals using only Riemann integration theory, avoiding Lebesgue integration.
• To address the case where the integrand $ f $ is bounded but not necessarily Riemann integrable, thus generalizing prior results.
• To clarify the relationship between Riemann–Stieltjes and Riemann integrals when $ G $ is an indefinite integral of a Riemann integrable function $ g $.

Proposed method

• Define the integrator $ G(x) = c + \int_a^x g(y)\,dy $, where $ g $ is Riemann integrable on $[a,b]$.
• Use the mixed Riemann sum estimate (Lemma 2.1) to bound the difference between Riemann–Stieltjes sums and the Riemann integral of $ fg $.
• Apply a sharp version of the mean value theorem for Riemann integrals (Theorem 1.2) to control the oscillation of $ g $ over subintervals.
• Establish equivalence between the convergence of Riemann–Stieltjes sums and the Riemann integrability of $ fg $, using oscillation bounds.
• Prove necessity by showing that if $ f $ is Riemann–Stieltjes integrable w.r.t. $ G $, then $ fg $ must be Riemann integrable, via norm-based estimates.
• Leverage the integration by parts formula for Riemann–Stieltjes integrals to derive symmetric integrability conditions (Corollary 3.2).

Experimental results

Research questions

• RQ1 Under what conditions is a bounded function $ f $ Riemann–Stieltjes integrable with respect to an integrator $ G $ that is an indefinite Riemann integral?
• RQ2 Can the Riemann–Stieltjes integral $ \int_a^b f\,dG $ be computed as a Riemann integral $ \int_a^b fg\,dx $ when $ G(x) = c + \int_a^x g(y)\,dy $?
• RQ3 How does the Riemann–Stieltjes integrability of $ f $ w.r.t. $ G $ relate to the Riemann integrability of the product $ fg $?
• RQ4 What is the minimal regularity required of $ G $ and $ f $ to ensure that the Riemann–Stieltjes integral reduces to a Riemann integral?
• RQ5 Can symmetric integrability conditions be derived when both $ f $ and $ G $ are defined via Riemann integrable densities?

Key findings

• A bounded function $ f $ is Riemann–Stieltjes integrable with respect to $ G(x) = c + \int_a^x g(y)\,dy $ if and only if the product $ fg $ is Riemann integrable on $[a,b]$.
• When the condition holds, the Riemann–Stieltjes integral satisfies $ \int_a^b f\,dG = \int_a^b fg\,dx $, enabling direct computation via Riemann integration.
• The proof remains entirely within Riemann integration theory, avoiding Lebesgue integration even when $ f $ is not Lebesgue measurable.
• The result generalizes the classical formula $ \int f\,dG = \int fG'\,dx $ to cases where $ G' = g $ is only Riemann integrable, not necessarily continuous.
• The necessity of $ fg $ being Riemann integrable is established via norm-based estimates on oscillation of $ g $, showing that failure of $ fg $'s integrability implies failure of Riemann–Stieltjes integrability.
• Corollary 3.2 establishes a symmetric integrability condition: if $ \alpha $ and $ \beta $ are Riemann integrable and one is an indefinite Riemann integral, then both $ \int \alpha\,d\beta $ and $ \int \beta\,d\alpha $ exist and equal Riemann integrals of the product.