Riemann–Stieltjes integral

In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the Lebesgue integral.

Definition

The Riemann–Stieltjes integral of a real-valued function f of a real variable with respect to a real function g is denoted by

$\int_a^b f(x) \, dg(x)$

and defined to be the limit, as the mesh of the partition

$P=\{ a = x_0 < x_1 < \cdots < x_n = b\}$

of the interval [a, b] approaches zero, of the approximating sum

$S(P,f,g) = \sum_{i=0}^{n-1} f(c_i)(g(x_{i+1})-g(x_i))$

where c_i is in the i-th subinterval [x_i, x_i+1]. The two functions f and g are respectively called the integrand and the integrator.

The "limit" is here understood to be a number A (the value of the Riemann–Stieltjes integral) such that for every ε > 0, there exists δ > 0 such that for every partition P with mesh(P) < δ, and for every choice of points c_i in [x_i, x_i+1],

$|S(P,f,g)-A| < \varepsilon. \,$

Generalized Riemann–Stieltjes integral

A slight generalization, introduced by Pollard (1920) and now standard in analysis, is to consider in the above definition partitions P that refine another partition P_ε, meaning that P arises from P_ε' by the addition of points, rather than from partitions with a finer mesh. Specifically, the generalized Riemann–Stieltjes integral of f with respect to g is a number A such that for every ε > 0 there exists a partition P_ε such that for every partition P that refines P_ε,

$|S(P,f,g) - A| < \varepsilon \,$

for every choice of points c_i in [x_i, x_i+1].

This generalization exhibits the Riemann–Stieltjes integral as the Moore–Smith limit on the directed set of partitions of [a, b] (McShane 1952). Hildebrandt (1938) calls it the Pollard–Moore–Stieltjes integral.

Darboux sums

The Riemann–Stieltjes integral can be efficiently handled using an appropriate generalization of Darboux sums. For a partition P and a nondecreasing function g on [a, b] define the upper Darboux sum of f with respect to g by

$U(P,f,g) = \sum_{i=1}^n \sup_{x\in [x_i,x_{i+1}]} f(x)\,\,(g(x_{i+1})-g(x_i))$

and the lower sum by

$L(P,f,g) = \sum_{i=1}^n \inf_{x\in [x_i,x_{i+1}]} f(x)\,\,(g(x_{i+1})-g(x_i)).$

Then the generalized Riemann–Stieltjes of f with respect to g exists if and only if, for every ε > 0, there exists a partition P such that

$U(P,f,g)-L(P,f,g) < \varepsilon.$

Furthermore, f is Riemann–Stieltjes integrable with respect to g (in the classical sense) if

$\lim_{\operatorname{mesh}(P)\to 0} [U(P,f,g)-L(P,f,g)] = 0.$

See Graves (1946, Chap. XII, §3).

Properties and relation to the Riemann integral

If g should happen to be everywhere differentiable, then the Riemann–Stieltjes integral may still be different from the Riemann integral of $f(x) g'(x)$ given by

$\int_a^b f(x) g'(x) \, dx,$

for example, if the derivative is unbounded. But if the derivative is continuous, they will be the same. This condition is also satisfied if g is the (Lebesgue) integral of its derivative; in this case g is said to be absolutely continuous.

However, g may have jump discontinuities, or may have derivative zero almost everywhere while still being continuous and increasing (for example, g could be the Cantor function), in either of which cases the Riemann–Stieltjes integral is not captured by any expression involving derivatives of g.

The Riemann–Stieltjes integral admits integration by parts in the form

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

and the existence of either integral implies the existence of the other (Hille & Phillips 1974, §3.3).

Existence of the integral

The best simple existence theorem states that if f is continuous and g is of bounded variation on [a, b], then the integral exists. A function g is of bounded variation if and only if it is the difference between two monotone functions. If g is not of bounded variation, then there will be continuous functions which cannot be integrated with respect to g. In general, the integral is not well-defined if f and g share any points of discontinuity, but this sufficient condition is not necessary.

On the other hand, a classical result of Young (1936) states that the integral is well-defined if f is α-Hölder continuous and g is β-Hölder continuous with α + β > 1.

Application to probability theory

If g is the cumulative probability distribution function of a random variable X that has a probability density function with respect to Lebesgue measure, and f is any function for which the expected value E(|f(X)|) is finite, then the probability density function of X is the derivative of g and we have

$E(f(X))=\int_{-\infty}^\infty f(x)g'(x)\, dx.$

But this formula does not work if X does not have a probability density function with respect to Lebesgue measure. In particular, it does not work if the distribution of X is discrete (i.e., all of the probability is accounted for by point-masses), and even if the cumulative distribution function g is continuous, it does not work if g fails to be absolutely continuous (again, the Cantor function may serve as an example of this failure). But the identity

$E(f(X))=\int_{-\infty}^\infty f(x)\, dg(x)$

holds if g is any cumulative probability distribution function on the real line, no matter how ill-behaved. In particular, no matter how ill-behaved the cumulative distribution function g of a random variable X, if the moment E(Xⁿ) exists, then it is equal to

$E(X^n) = \int_{-\infty}^\infty x^n\,dg(x).$

Application to functional analysis

The Riemann–Stieltjes integral appears in the original formulation of F. Riesz's theorem which represents the dual space of the Banach space C[a,b] of continuous functions in an interval [a,b] as Riemann–Stieltjes integrals against functions of bounded variation. Later, that theorem was reformulated in terms of measures.

The Riemann–Stieltjes integral also appears in the formulation of the spectral theorem for (non-compact) self-adjoint (or more generally, normal) operators in a Hilbert space. In this theorem, the integral is considered with respect to a spectral family of projections. See Riesz & Sz. Nagy (1955) for details.

Generalization

An important generalization is the Lebesgue–Stieltjes integral which generalizes the Riemann–Stieltjes integral in a way analogous to how the Lebesgue integral generalizes the Riemann integral. If improper Riemann–Stieltjes integrals are allowed, the Lebesgue integral is not strictly more general than the Riemann–Stieltjes integral.

The Riemann–Stieltjes integral also generalizes to the case when either the integrand ƒ or the integrator g take values in a Banach space. If g : [a,b] → X takes values in the Banach space X, then it is natural to assume that it is of strongly bounded variation, meaning that

$\sup \sum_i \|g(t_i)-g(t_{i+1})\|_X < \infty$

the supremum being taken over all finite partitions

$a=t_0\le t_1\le\cdots\le t_n=b$

of the interval [a,b]. This generalization plays a role in the study of semigroups, via the Laplace–Stieltjes transform.

References

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