The Riemann–Stieltjes Integral — Mathematics Knowledge System

Overview

The Riemann–Stieltjes integral generalizes the Riemann integral by replacing the increment (xᵢ₊₁ - xᵢ) of the integration variable with the increment (α(xᵢ₊₁) - α(xᵢ)) of a second function, the integrator α. Written ∫ₐᵇ f dα, it reduces to the ordinary Riemann integral when α(x)=x, but allows integrating against functions with jumps, kinks, or other non-smooth behavior — in particular, when α is a step function, the Riemann–Stieltjes integral collapses to a weighted sum evaluated exactly at the jump points, unifying discrete sums and continuous integrals into a single framework. Developed by Thomas Joannes Stieltjes in 1894 while studying continued fractions, it became foundational for probability theory (integrating against a cumulative distribution function handles discrete, continuous, and mixed random variables uniformly) and for the later, more general Lebesgue–Stieltjes integral.

Intuition

In the ordinary Riemann integral, each rectangle's width is the plain length of a sub-interval, (xᵢ₊₁-xᵢ), treating every point of the x-axis as contributing equally. In the Riemann–Stieltjes integral, the 'width' of each rectangle is instead reweighted by how much the integrator α changes across that sub-interval, α(xᵢ₊₁)-α(xᵢ). If α increases steeply somewhere, that region of the x-axis is given outsized importance; if α is locally constant, that region contributes nothing at all, no matter how wide it is. When α has a sudden jump at some point c, all of the 'weight' near c concentrates onto that single point, and the integral simply picks out f(c) times the jump size — turning the integral, in the limit, into exactly the kind of discrete term you'd expect from a sum, not an area.

Formal Definition

Definition

Let f be bounded and α be monotonically increasing (or, more generally, of bounded variation) on [a,b]. For a tagged partition a=x₀<x₁<⋯<xₙ=b with tags tᵢ∈[xᵢ,xᵢ₊₁], the Riemann–Stieltjes sum is:

\sum_{i=0}^{n-1} f(t_i)\,\big[\alpha(x_{i+1}) - \alpha(x_i)\big]

Riemann–Stieltjes sum

\forall \varepsilon>0\ \exists \delta>0 \text{ such that mesh}(P)<\delta \implies \left|\sum_{i} f(t_i)[\alpha(x_{i+1})-\alpha(x_i)] - s\right| < \varepsilon

f is RS-integrable with respect to α, with integral s, if sufficiently fine partitions force every tagged sum within ε of s regardless of tag choice

Definition of the Riemann–Stieltjes integral

s = \int_a^b f(x)\,d\alpha(x)

Notation for the Riemann–Stieltjes integral of f with respect to α

\alpha(x) = x \implies \int_a^b f\,d\alpha = \int_a^b f(x)\,dx

When the integrator is the identity function, the RS integral is exactly the classical Riemann integral

Reduces to the ordinary Riemann integral

Notation

Notation	Meaning
$\int_a^b f(x)\,d\alpha(x)$	The Riemann–Stieltjes integral of f with respect to the integrator α over [a,b]
$\alpha$	The integrator function; typically taken monotonically increasing, or more generally of bounded variation
$\Delta \alpha_i = \alpha(x_{i+1}) - \alpha(x_i)$	The increment of the integrator across the i-th sub-interval, replacing the plain width Δxᵢ used in the Riemann integral

Derivation

Sketch of why, when α is a pure step function with a single jump of size c at x₀, the Riemann–Stieltjes integral of a continuous f reduces exactly to f(x₀)·c.

\alpha(x) = \begin{cases} 0 & x < x_0 \\ c & x \ge x_0 \end{cases}

A single-jump step function integrator

\alpha(x_{i+1}) - \alpha(x_i) = 0 \text{ for every sub-interval not straddling } x_0

α is constant away from x₀, so those terms of the Riemann–Stieltjes sum vanish entirely

\alpha(x_{i+1}) - \alpha(x_i) = c \text{ for the (unique) sub-interval containing } x_0

Only the sub-interval that straddles the jump contributes a nonzero increment, equal to the full jump size c

\text{Sum} = f(t_i)\cdot c \ \xrightarrow{\text{mesh}\to 0} \ f(x_0)\cdot c

As the partition is refined, the straddling sub-interval shrinks around x₀, and continuity of f forces f(t_i)\to f(x_0)

Properties

Reduces to Riemann integral

\alpha(x) = x \implies \int_a^b f\,d\alpha = \int_a^b f\,dx

Linearity in f

\int_a^b (c_1 f_1 + c_2 f_2)\,d\alpha = c_1\int_a^b f_1\,d\alpha + c_2 \int_a^b f_2\,d\alpha

Linearity in α

\int_a^b f\,d(c_1\alpha_1+c_2\alpha_2) = c_1\int_a^b f\,d\alpha_1 + c_2\int_a^b f\,d\alpha_2

Integration by parts (Stieltjes)

\int_a^b f\,d\alpha + \int_a^b \alpha\,df = f(b)\alpha(b) - f(a)\alpha(a)

Condition: Holds whenever one of the two integrals exists; a hallmark identity with no direct analogue for the plain Riemann integral.

Existence criterion

\text{If } f \text{ is continuous on } [a,b] \text{ and } \alpha \text{ is of bounded variation, then } \int_a^b f\,d\alpha \text{ exists.}

Condition: More generally, existence holds whenever f and α have no common points of discontinuity.

Jump reduction

\text{If } \alpha \text{ has an isolated jump of size } c \text{ at } x_0 \in (a,b) \text{ and } f \text{ is continuous at } x_0, \text{ then the contribution to } \int f\,d\alpha \text{ from that jump is } f(x_0)\cdot c.

Condition: This is the mechanism that lets Riemann–Stieltjes integration reproduce discrete sums exactly.

Applications

Expected value and pricing formulas in probability write E[g(X)] = ∫g(x) dF(x), where F is the cumulative distribution function of X — a Riemann–Stieltjes (or Lebesgue–Stieltjes) integral that handles discrete random variables (F a step function, integral becomes a sum), continuous random variables (F differentiable, integral becomes ∫g(x)f(x)dx), and mixed distributions all within one formula.

Worked Examples

Since f(x)=x² is continuous and α is a step function with isolated jumps of size 1 at x=1 and x=2, each jump contributes f(jump point) times jump size.
$\int_0^3 x^2\,d\alpha(x) = f(1)\cdot 1 + f(2)\cdot 1$
Evaluate f at each jump point: f(1)=1²=1, f(2)=2²=4.
$f(1) = 1, \quad f(2) = 4$
Sum the contributions.
$\int_0^3 x^2\,d\alpha(x) = 1 + 4 = 5$

Answer: 5

Practice Problems

Difficulty 6/10

Let α(x) jump by 3 at x=2 only (and is otherwise constant) on [0,4]. Compute ∫₀⁴ (x+1) dα(x) for f(x)=x+1.

Difficulty 6/10

If α(x) = x for all x, then ∫ₐᵇ f dα equals:

0, alwaysThe ordinary Riemann integral ∫ₐᵇ f(x) dxf(b) - f(a)It is undefined

Difficulty 7/10

A discrete random variable X takes value 2 with probability 0.3 and value 5 with probability 0.7. Using F (the CDF, a step function with jumps 0.3 at x=2 and 0.7 at x=5) and the Riemann-Stieltjes jump-reduction property, compute E[X] as the sum of x times each jump in F.

Common Mistakes

Common Mistake

Assuming the Riemann–Stieltjes integral always requires α to be differentiable, so that dα = α'(x)dx converts it back to an ordinary integral.

The entire point of the RS integral is to handle integrators that are NOT differentiable — e.g. step functions with jumps. When α is differentiable with continuous α', it's true that ∫f dα = ∫f(x)α'(x)dx, but the RS integral is defined and useful far beyond that special case.

Common Mistake

Believing ∫f dα only depends on f, ignoring that swapping the integrator α changes the answer entirely.

The integrator α determines how much 'weight' each region of [a,b] receives; the same f integrated against two different integrators (e.g. α(x)=x versus a step function) generally gives completely different results.

Common Mistake

Thinking that if f and α are both discontinuous at some shared point, the integral can still be salvaged by choosing convenient tag points.

If f and α share a discontinuity point, the Riemann–Stieltjes sums can be made to converge to different values depending on tag choice near that point, so the integral genuinely fails to exist — no choice of tags fixes this.

Quiz

Flashcards

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Summary

The Riemann–Stieltjes integral ∫f dα generalizes the Riemann integral by replacing the sub-interval width (xᵢ₊₁-xᵢ) with the integrator's increment α(xᵢ₊₁)-α(xᵢ).
When α(x)=x, it reduces exactly to the ordinary Riemann integral.
When α is a step function, isolated jumps of size c at x₀ contribute f(x₀)·c to the integral — turning integrals into sums, and unifying discrete and continuous settings.
Existence is guaranteed when f is continuous and α is of bounded variation (or more generally, f and α share no common discontinuity).
The framework underlies E[g(X)] = ∫g dF in probability, letting one formula handle discrete, continuous, and mixed random variables via their CDF F.

References

WebsiteWikipedia — Riemann–Stieltjes integral
BookRudin, W. Principles of Mathematical Analysis, 3rd ed. Ch. 6.