Stieltjes Integral

The Stieltjes integral is a generalization of the Riemann Integral. Let $f(x)$ and $\alpha(x)$ be real-values bounded functions defined on a Closed Interval $[a,b]$. Take a partition of the Interval

$$a=x_0<x_1<x_2,\ldots<x_{n-1}<x_n=b,$$ (1)

and consider the Riemann sum

$$\sum_{i=0}^{n-1} f(\xi_i)[\alpha(x_{i+1})-\alpha(x_i)]$$ (2)

with $\xi_i\in[x_i,x_{i+1}]$. If the sum tends to a fixed number $I$ as $\max(x_{i+1}-x_i)\to 0$, then $I$ is called the Stieltjes integral, or sometimes the Riemann-Stieltjes Integral. The Stieltjes integral of $f$ with respect to $\alpha$ is denoted

$$\int f(x)\,d\alpha(x)$$ (3)

or sometimes simply

$$\int f\,d\alpha.$$ (4)

If $f$ and $\alpha$ have a common point of discontinuity, then the integral does not exist. However, if the Stieltjes integral exists and $f$ has a Derivative $f'$, then

$$\int f(x)\,d\alpha(x)=\int f(x)\alpha'(x)\,dx.$$ (5)

For enumeration of many of the integral's properties, see Dresher (1981, p. 105).

See also Riemann Integral

References

Dresher, M. The Mathematics of Games of Strategy: Theory and Applications. New York: Dover, 1981.

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 152-155, 1988.

Kestelman, H. ``Riemann-Stieltjes Integration.'' Ch. 11 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 247-269, 1960.