NOTE ON THE FIRST FUNDAMENTAL THEOREM

FOR RIEMANN–STIELTJES INTEGRALS

Robert Cacioppo

Northeast Missouri State University

A Riemann–Stieltjes integral is of the form ∫_{a}^{b} fdα. The functions f and α are called the integrand and integrator, respectively. If the integral exists, we say f ∈ R(α) on [a, b]. We will always assume the integrator is of bounded variation (a sufficient condition for this is that α be monotonic, although this is not necessary). This insures that if f ∈ R(α) on [a, b] then f ∈ R(α) on [a, x] also, for all x ∈ [a, b]. Thus the function F (x) = ∫_{a}^{x} fdα is well–defined on [a, b]. The special case of the Riemann integral occurs when α(x) = x.

The first fundamental theorem of calculus for the Riemann–Stieltjes integral applies when the integrator is monotonic on [a, b]. In this case,

\frac{d}{dx} ∫_{a}^{x} fdα = f(x)α′(x)

for each x ∈ (a, b) at which f is continuous and α′ exists. Thus for Riemann integrals, the only requirement is the continuity of f at x.

The requirement that α be monotonic can be dropped if α′ is continuous on [a, b]. In this case, the Riemann–Stieltjes integral reduces to a Riemann integral: ∫_{a}^{x} fdα = ∫_{a}^{x} fα′. Thus the only requirement for the differentiability of this integral at x is the continuity of f at x. For proofs of these results, see [1].

The fundamental theorem can be easily extended to include all integrators α of bounded variation on the interval [a, b] for which

(1) α′ exists on (a, b), and

(2) for each x ∈ (a, b), there is a neighborhood of x on which α′ is bounded. For instance, the fundamental theorem applies, even at x = 0, with the integrator

α(x) = \{ \begin{matrix}x^{2} sin \frac{1}{x}, & if x ≠ 0; \\ 0, & if x = 0. \end{matrix}

To prove the extended version, assume for some neighborhood of x that α = β +γ where β and γ are monotonic and differentiable at x. The extended version now follows from the monotonic version, using the linearity of integration with respect to a sum of integrators.

To complete the proof, given x ∈ (a, b), choose a neighborhood B of x and constant M such that M ≥ |α′(t)| for all t ∈ B. On B, define β(t) = Mt and γ = α − β. Now γ is decreasing on B since if t_{1} < t_{2} then, by the Mean Value Theorem [1],

\begin{aligned}γ(t_{1}) − γ(t_{2}) & = α′(c)(t_{1} − t_{2}) − M(t_{1} − t_{2}) \\ & = (α′(c) − M)(t_{1} − t_{2}) ≥ 0 , \end{aligned}

for some c ∈ (t_{1}, t_{2}).

\underline{Reference}

1. T. M. Apostol, Mathematical Analysis, 2nd edition, Addison–Wesley, 1975.