Riemann–Stieltjes Integration: Theory & Applications

• Riemann–Stieltjes integration is a generalization of the classical Riemann integral that integrates with respect to functions of bounded variation.
• It underpins applications in stochastic calculus, numerical quadrature, time scales analysis, and integration on Banach spaces.
• Recent advances establish robust existence criteria, precise error estimates, and innovative variable substitution techniques across diverse mathematical frameworks.

Riemann–Stieltjes integration generalizes the classical Riemann integral by allowing integration with respect to a general function of bounded variation rather than the identity function. This concept appears in numerous contexts: analysis, probability theory, stochastic calculus, fractional processes, time scales, and integration on algebraic structures. The Riemann–Stieltjes integral is defined through sums involving increments of the integrator and admits key generalizations, error estimates, and foundational theorems on existence, convergence, and numerical approximation.

1. Classical Definition, Generalization, and Key Properties

Let $[a,b] \subseteq \mathbb{R}$, $f$ a bounded function, and $g$ a function of bounded variation. The Riemann–Stieltjes integral $\int_a^b f(x) dg(x)$ is defined via partition sums

$$S(P, f, g) = \sum_{i=1}^n f(\xi_i) [g(x_i) - g(x_{i-1})]$$

where $P = \{a = x_0 < x_1 < \dots < x_n = b\}$; the sample points $\xi_i \in [x_{i-1}, x_i]$. If the sums converge as the mesh of $P$ tends to zero independent of the choice of $\xi_i$, the function $f$ is said to be Riemann–Stieltjes integrable with respect to $g$.

Key classical properties:

• Linearity: Integration is linear in both the integrand and integrator.
• Additivity: Integrals over adjacent intervals sum.
• Integration by parts: If both $f$ and $g$ are of bounded variation,

$$\int_a^b f(x)\,dg(x) = f(b)g(b) - f(a)g(a) - \int_a^b g(x)\,df(x)$$

• Criteria for existence: Sufficient regularity (e.g., continuous $f$, $g$ of bounded variation) ensures existence.

A central result extends the theory to the case where the integrator $G$ is given as an indefinite Riemann integral $G(x) = c + \int_a^x g(y) dy$, showing that $\int_a^b f(x)\,dG(x) = \int_a^b f(x)g(x)\,dx$ provided $f g$ is Riemann integrable—even if $f$ itself is not (Pouso, 2011).

2. Generalizations: Time Scales, Banach Spaces, Hyperbolic Numbers, and Algebraic Categories

a. Time Scales

The Riemann–Stieltjes integral framework is lifted to arbitrary time scales $T$ (unifying discrete, continuous, and quantum calculus), using partitions adapted to the structure ("delta" or "nabla" intervals), with Darboux–Stieltjes sums

$$U(P, f, g) = \sum_{j=1}^n \sup_{t \in [t_{j-1}, t_j]_{\square}} f(t) (g(t_j) - g(t_{j-1}))$$

$$L(P, f, g) = \sum_{j=1}^n \inf_{t \in [t_{j-1}, t_j]_{\square}} f(t) (g(t_j) - g(t_{j-1}))$$

where $\square$ refers to delta or nabla sense (0903.1224). Integrability and classical properties transport with only structural modification. For $T = \mathbb{Z}$, every function is integrable and the integral becomes

$$\int_a^b f(t) \square g(t) = \sum_{t=a}^{b-1} f(t)[g(t+1) - g(t)]$$

Extensions include majorisation inequalities, Jensen-type and Fubini-type (double integral) generalizations (Mozyrska et al., 2010).

b. Banach Spaces and Regulated/Irregular Signals

For functions and integrators valued in Banach spaces, the existence and estimation theory leverages truncated variation

$$TV^\delta(f; [a, b]) = \sup_P \sum_{i} \max\{|f(t_i)-f(t_{i-1})|-\delta, 0\}$$

and introduces spaces $\mathcal{U}^p([a,b], W)$—"regulated signals"* (Editor's term)—consisting of functions approximable (in uniform norm) by functions with total variation scaling as $\delta^{1-p}$ (Łochowski, 2016). For any $f$ and $g$ in $\mathcal{U}^p$ and $\mathcal{U}^q$, an improved Loéve–Young inequality applies: $$\left| \int_a^b f\,dg - f(a)[g(b)-g(a)] \right| \leq C_{p,q}\ \|f\|_{p-\text{var}}^{1-1/q}\ \|f\|_{\operatorname{osc}}^{1+ p/q - p}\ \|g\|_{q-\text{var}}^{1/q}$$ Rate-independent conditions admit highly irregular signals as integrators.

c. Hyperbolic Numbers and Algebraic Integration

On hyperbolic intervals, integration uses strong partitions with idempotent representation, so hyperbolic integrals reduce component-wise to classical integrals (Tellez-Sanchez et al., 2021). In finite-dimensional algebras, a mapping $\omega$ (monotone, absolutely continuous, bijective, or identity) determines whether the integration framework recovers Lebesgue–Stieltjes, Riemann–Stieltjes, or substitution formulas (Gao et al., 31 May 2024). Categorically, the main formula is

$$\int_{I_1} f\,d(\varphi \circ p_1) = \int_{I_2} (f \circ \omega)\,dp_2$$

with the precise properties of $\omega$ dictating the measure-theoretic or Riemannian nature of the integration.

3. Inequalities, Error Estimates, and Quadrature

A substantial number of papers develop sharp inequalities, error bounds, and quadrature formulas:

• Weighted Ostrowski–generalized trapezoid and Simpson-type inequalities, parameterized by $\alpha$:

$$\left| \alpha[(u(x) - u(a))f(a) + (u(b) - u(x))f(b)] + (1-\alpha)\{[u(\tfrac{a+b}{2}) - u(a)]f(x) + [u(b) - u(\tfrac{a+b}{2})]f(a+b-x)\} - \int_a^b f du \right| \leq (\dots) V_a^b(f)$$

with optimal constants (Alomari, 2014).

• Gauss–Legendre quadrature rules for Riemann–Stieltjes integrals, e.g., for $f$ of Hölder regularity and $g$ of bounded variation or Lipschitz,

$$\int_{-1}^1 f(t)\,dg(t) \approx A f(-\tfrac{\sqrt{3}}{3}) + B f(\tfrac{\sqrt{3}}{3})$$

where $A$, $B$ depend explicitly on $g$ (Alomari, 2014).

• General $L^p$ error estimates for two- and three-point rules, using new "triangle" inequalities:

$$|\int_a^b w(t) dv(t)| \leq L (b-a)^{1 - 1/p} \|w\|_p$$

with $L$ the Lipschitz constant of $v$ (Alomari et al., 2018).

These advances enable robust estimation of approximation errors in numerical integration, with quantifiable dependence on the variation and regularity of the functions involved.

4. Existence, Convergence, and Change of Variables

General existence criteria unify classical and modern approaches:

• Truncated variation theorem: If the series $$S = \sum_k [2^k n_{k-1} TV^{\theta_k}(g) + 2^k \theta_k TV^{n_k}(f)]$$ is finite, $\int_a^b f\,dg$ exists and $| \int f\,dg - f(a)[g(b) - g(a)]| < S$ (Łochowski, 2014).
• Bounded convergence theorem: For regulated sequences in Banach spaces converging pointwise and bounded, integration commutes with the limit (Monteiro et al., 2014).
• Change of variable/substitution: If $f$ is integrable w.r.t. $Y$ on $\varphi([a, b])$, $$\int_{\varphi(a)}^{\varphi(b)} f(y)\,dY(y) = \int_a^b f(\varphi(x))\,d(Y \circ \varphi)(x)$$ even if $\varphi$ is non-invertible or monotone (Torchinsky, 2019).

This significantly broadens the class of admissible integrators and integrands, including functions with discontinuities or highly irregular paths, and enables integration in measure-theoretic or more algebraic settings.

5. Applications: Stochastic Integration, Fractional Processes, PDEs, and Physical Modeling

• Fractional Brownian motion: Sufficient conditions for existence of $\int_a^b Y\,dX$ are given in terms of two-parameter control functions, circumventing restrictions due to unbounded $p$-variation (e.g., for indicator functions composed with fBm) (Yaskov, 2015).
• Stochastic integration: Pathwise Stieltjes integrals are well-defined for discontinuously evaluated processes, where the evaluation function $f$ may have locally infinite variation and the driving process $X$ is sufficiently "variable" in the sense of density bounds; fractional calculus approaches (Zähle–Stieltjes integral) yield convergence rates for numerical approximations (Chen et al., 2016).
• Sustainability modeling: In global sustainability indices, Riemann–Stieltjes integrals are used to weight multivariate factors via associated weight functions, tightly coupling the PDE solutions with variable-specific importance, allowing precise and adaptable quantitative indices (e.g. in civil engineering for pavement design) (Rao et al., 2021).
• Signal processing and physics: Modified Riemann sums, where each partition interval is locally shrunk, still converge to the classical integral, validating practical integration schemes in experimental setups or amplitude-modulated signal reconstruction (Torchinsky, 2019).

6. Comparison to Lebesgue–Stieltjes Integration and Measure-Theoretic Frameworks

Riemann–Stieltjes and Lebesgue–Stieltjes integrals coincide under suitable conditions:

• For discrete distribution functions $F$ that are constant on intervals and jump at finitely many points, both integration frameworks yield the same value provided the integrand $f$ is continuous at the points of discontinuity (Niang et al., 2020).
• For Lebesgue–Stieltjes, $F$ must be right-continuous and non-decreasing; the induced measure (by Carathéodory's extension theorem) naturally handles infinite or countable sets of discontinuities. Riemann–Stieltjes is more computationally direct but less robust in the presence of dense jumps.
• In categorical algebraic frameworks, Lebesgue–Stieltjes and Riemann–Stieltjes integration are both subsumed by normed-module integrals, with the nature of the mapping $\omega$ (monotonicity, absolute continuity, bijection) selecting the appropriate classical theory (Gao et al., 31 May 2024).

7. Advanced Theorems and Modern Directions

• Improvements to classical inequalities via oscillation norms and truncated variation facilitate integration with highly irregular signals (improved Loéve–Young inequalities).
• The “rate-independent” characterization enables integration of functions lacking finite $p$-variation yet possessing controllable truncated variation rates.
• In rough path theory and stochastic analysis, these results underpin existence and quantitative estimates for integral equations driven by rough, non-semimartingale signals, extending the foundational work of Terry Lyons (Łochowski, 2014, Łochowski, 2016).

Key Formulas

• Upper and lower Darboux–Stieltjes sums (time scales):

$$U(P, f, g) = \sum_{j=1}^n \sup\{f(t): t \in [t_{j-1}, t_j]_{\square}\} (g(t_j) - g(t_{j-1}))$$

$$L(P, f, g) = \sum_{j=1}^n \inf\{f(t): t \in [t_{j-1}, t_j]_{\square}\} (g(t_j) - g(t_{j-1}))$$

• Integration by parts:

$$\int_a^b f(t)\,dg(t) = [f(t)g(t)]_a^b - \int_a^b g(t)\,df(t)$$

• Improved Loéve–Young inequality:

$$|\int_a^b f\,dg - f(a)[g(b)-g(a)]| \leq C_{p,q} (V_p(f))^{1-1/q} \|f\|_{\operatorname{osc}}^{1+p/q-1} (V_q(g))^{1/q}$$

• Change of variables (noninvertible):

$$\int_{\varphi(a)}^{\varphi(b)} f(y) dY(y) = \int_a^b f(\varphi(x)) d(Y \circ \varphi)(x)$$

Table: Comparison of Riemann–Stieltjes and Lebesgue–Stieltjes Integration

Feature	Riemann–Stieltjes	Lebesgue–Stieltjes
Integrator regularity	Bounded variation	Right-continuous, nondecreasing
Handling of discontinuities	f must be continuous at jumps	Measure handles jumps automatically
Computational approach	Sums over partitions	Integration with respect to a measure
Infinite jumps	Cumbersome	Naturally incorporated
Coincidence conditions	f continuous at jumps	f measurable, measure is proper

Summary

Riemann–Stieltjes integration plays a foundational role in analysis and applications requiring integration against functions of bounded variation, notably in stochastic processes, rough path analysis, numerical integration, the theory on arbitrary time scales, and on algebraic structures. Modern developments refine classical existence and error bounds, extend applicability to irregular signals and multidimensional contexts, establish deep connections with Lebesgue–Stieltjes integration, and provide generalization and unification in advanced mathematical frameworks.