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 realvalued function f of a real variable with respect to a real function g is denoted by
 $\backslash int\_a^b\; f(x)\; \backslash ,\; dg(x)$
and defined to be the limit, as the mesh of the partition
 $P=\backslash \{\; a\; =\; x\_0\; <\; x\_1\; <\; \backslash cdots\; <\; x\_n\; =\; b\backslash \}$
of the interval [a, b] approaches zero, of the approximating sum
 $S(P,f,g)\; =\; \backslash sum\_\{i=0\}^\{n1\}\; f(c\_i)(g(x\_\{i+1\})g(x\_i))$
where c_{i} is in the ith 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}],
 $S(P,f,g)A\; <\; \backslash varepsilon.\; \backslash ,$
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},
 $S(P,f,g)\; \; A\; <\; \backslash varepsilon\; \backslash ,$
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
 $U(P,f,g)\; =\; \backslash sum\_\{i=1\}^n\; \backslash sup\_\{x\backslash in\; [x\_i,x\_\{i+1\}]\}\; f(x)\backslash ,\backslash ,(g(x\_\{i+1\})g(x\_i))$
and the lower sum by
 $L(P,f,g)\; =\; \backslash sum\_\{i=1\}^n\; \backslash inf\_\{x\backslash in\; [x\_i,x\_\{i+1\}]\}\; f(x)\backslash ,\backslash ,(g(x\_\{i+1\})g(x\_i)).$
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
 $U(P,f,g)L(P,f,g)\; <\; \backslash varepsilon.$
Furthermore, f is Riemann–Stieltjes integrable with respect to g (in the classical sense) if
 $\backslash lim\_\{\backslash operatorname\{mesh\}(P)\backslash to\; 0\}\; [U(P,f,g)L(P,f,g)]\; =\; 0.$
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 $f(x)\; g\text{'}(x)$ given by
 $\backslash int\_a^b\; f(x)\; g\text{'}(x)\; \backslash ,\; dx,$
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
 $\backslash int\_a^b\; f(x)\; \backslash ,\; dg(x)=f(b)g(b)f(a)g(a)\backslash int\_a^b\; g(x)\; \backslash ,\; df(x).$
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 welldefined 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 welldefined if f is αHölder continuous and g is βHölder 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
 $E(f(X))=\backslash int\_\{\backslash infty\}^\backslash infty\; f(x)g\text{'}(x)\backslash ,\; dx.$
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 pointmasses), 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
 $E(f(X))=\backslash int\_\{\backslash infty\}^\backslash infty\; f(x)\backslash ,\; dg(x)$
holds if g is any cumulative probability distribution function on the real line, no matter how illbehaved. In particular, no matter how illbehaved the cumulative distribution function g of a random variable X, if the moment E(X^{n}) exists, then it is equal to
 $E(X^n)\; =\; \backslash int\_\{\backslash infty\}^\backslash infty\; x^n\backslash ,dg(x).$
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 (noncompact) selfadjoint (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
 $\backslash sup\; \backslash sum\_i\; \backslash g(t\_i)g(t\_\{i+1\})\backslash \_X\; <\; \backslash infty$
the supremum being taken over all finite partitions
 $a=t\_0\backslash le\; t\_1\backslash le\backslash cdots\backslash le\; t\_n=b$
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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 Methods  

 Improper Integrals  

 Stochastic integrals  


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