## Stieltjes Integral

The Stieltjes integral is a generalization of the Riemann Integral. Let and be real-values bounded functions defined on a Closed Interval . Take a partition of the Interval

 (1)

and consider the Riemann sum
 (2)

with . If the sum tends to a fixed number as , then is called the Stieltjes integral, or sometimes the Riemann-Stieltjes Integral. The Stieltjes integral of with respect to is denoted
 (3)

or sometimes simply
 (4)

If and have a common point of discontinuity, then the integral does not exist. However, if the Stieltjes integral exists and has a Derivative , then
 (5)

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

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.