Riemann–Stieltjes integral

The Riemann–Stieltjes integral of a real-valued function ${\displaystyle f}$ of a real variable on the interval ${\displaystyle [a,b]}$ with respect to another real-to-real function ${\displaystyle g}$ is denoted by

${\displaystyle \int _{x=a}^{b}f(x)\,\mathrm {d} g(x).}$

Its definition uses a sequence of partitions ${\displaystyle P}$ of the interval ${\displaystyle [a,b]}$

${\displaystyle P=\{a=x_{0}<x_{1}<\cdots <x_{n}=b\}.}$

The integral, then, is defined to be the limit, as the mesh (the length of the longest subinterval) of the partitions approaches ${\displaystyle 0}$, of the approximating sum

${\displaystyle S(P,f,g)=\sum _{i=0}^{n-1}f(c_{i})\left[g(x_{i+1})-g(x_{i})\right]}$

where ${\displaystyle c_{i}}$ is in the ${\displaystyle i}$-th subinterval ${\displaystyle [x_{i};x_{i+1}]}$. The two functions ${\displaystyle f}$ and ${\displaystyle g}$ are respectively called the integrand and the integrator. Typically ${\displaystyle g}$ is taken to be monotone (or at least of bounded variation) and right-semicontinuous (however this last is essentially convention). We specifically do not require ${\displaystyle g}$ to be continuous, which allows for integrals that have point mass terms.

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 norm(P) < δ, and for every choice of points c_i in [x_i, x_i+1],

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

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

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} g(x)=f(b)g(b)-f(a)g(a)-\int _{a}^{b}g(x)\,\mathrm {d} f(x)}$

and the existence of either integral implies the existence of the other.[2]

On the other hand, a classical result[3] shows that the integral is well-defined if f is α-Hölder continuous and g is β-Hölder continuous with α + β > 1 .

If ${\displaystyle f(x)}$ is bounded on ${\displaystyle [a,b]}$, ${\displaystyle g(x)}$ increases monotonically, and ${\displaystyle g'(x)}$ is Riemann integrable, then the Riemann–Stieltjes integral is related to the Riemann integral by

$${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} g(x)=\int _{a}^{b}f(x)g'(x)\,\mathrm {d} x}$$

For a step function

$${\displaystyle g(x)={\begin{cases}0&{\text{if }}x\leq s\\1&{\text{if }}x>s\\\end{cases}}}$$

where ${\displaystyle a<s<b}$, if ${\displaystyle f}$ is continuous at ${\displaystyle s}$, then

$${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} g(x)=f(s)}$$

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 ${\displaystyle \operatorname {E} \left[\,\left|f(X)\right|\,\right]}$ is finite, then the probability density function of X is the derivative of g and we have

${\displaystyle \operatorname {E} [f(X)]=\int _{-\infty }^{\infty }f(x)g'(x)\,\mathrm {d} x.}$

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

${\displaystyle \operatorname {E} [f(X)]=\int _{-\infty }^{\infty }f(x)\,\mathrm {d} g(x)}$

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

${\displaystyle \operatorname {E} \left[X^{n}\right]=\int _{-\infty }^{\infty }x^{n}\,\mathrm {d} g(x).}$

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.[4]

The best simple existence theorem states that if f is continuous and g is of bounded variation on [a, b], then the integral exists.[5][6][7] A function g is of bounded variation if and only if it is the difference between two (bounded) 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 there are other cases as well.

A 3D plot, with ${\displaystyle x}$, ${\displaystyle f(x)}$, and ${\displaystyle g(x)}$ all along orthogonal axes, leads to a geometric interpretation of the Riemann–Stieltjes integral.[8]

The basic geometry of the Riemann-Stieljes integral.

If the ${\displaystyle g}$-${\displaystyle x}$ plane is horizontal and the ${\displaystyle f}$-direction is pointing upward, then the surface to be considered is like a curved fence. The fence follows the curve traced by ${\displaystyle g(x)}$, and the height of the fence is given by ${\displaystyle f(x)}$. The fence is the section of the ${\displaystyle g}$-sheet (i.e., the ${\displaystyle g}$ curve extended along the ${\displaystyle f}$ axis) that is bounded between the ${\displaystyle g}$-${\displaystyle x}$ plane and the ${\displaystyle f}$-sheet. The Riemann-Stieljes integral is the area of the projection of this fence onto the ${\displaystyle f}$-${\displaystyle g}$ plane — in effect, its "shadow".

The slope of ${\displaystyle g(x)}$ weights the area of the projection. The values of ${\displaystyle x}$ for which ${\displaystyle g(x)}$ has the steepest slope ${\displaystyle g'(x)}$ correspond to regions of the fence with the greater projection and thereby carry the most weight in the integral.

The effects of curvature in ${\displaystyle g(x)}$ on the geometry of the Riemann-Stieljes integral.

When ${\displaystyle g}$ is a step function

$${\displaystyle g(x)={\begin{cases}0&{\text{if }}x\leq s\\1&{\text{if }}x>s\\\end{cases}}}$$

the fence has a rectangular "gate" of width 1 and height equal to ${\displaystyle f(s)}$. Thus the gate, and its projection, have area equal to ${\displaystyle f(s)}$, the value of the Riemann-Stieljes integral.

The effect of a step function ${\displaystyle g(x)}$ on the geometry of the Riemann-Stieljes integral.

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, then 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

${\displaystyle \sup \sum _{i}\|g(t_{i-1})-g(t_{i})\|_{X}<\infty }$

the supremum being taken over all finite partitions

${\displaystyle a=t_{0}\leq t_{1}\leq \cdots \leq t_{n}=b}$

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

The Itô integral extends the Riemann–Stietjes integral to encompass integrands and integrators which are stochastic processes rather than simple functions; see also stochastic calculus.

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