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Stieltjes Integral

The Stieltjes integral is a generalization of the Riemann integral. Let f(x) and alpha(x) be real-valued bounded functions defined on a closed interval [a,b]. Take a partition of the interval

a=x_0<x_1<x_2,...<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)->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

intf(x)dalpha(x)
(3)

or sometimes simply

intfdalpha.
(4)

If f and alpha have a common point of discontinuity, then the integral does not exist. However, if f is continuous and alpha^' is Riemann integrable over the specified interval, then

intf(x)dalpha(x)=intf(x)alpha^'(x)dx
(5)

(Kestelman 1960).

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

See also

Convolution, Riemann Integral

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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.Jeffreys, H. and Jeffreys, B. S. "Integration: Riemann, Stieltjes." §1.10 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 26-36, 1988.Kestelman, H. "Riemann-Stieltjes Integration." Ch. 11 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 247-269, 1960.Pollard, S. Quart. J. Math. 49, 73-138, 1923.Stieltjes, T. J. "Recherches sur les fractions continues." Ann. d. fac. d. sciences Toulouse 8, No. 4, J1-J122, 1894.Widder, D. V. Ch. 1 in The Laplace Transform. Princeton, NJ: Princeton University Press, 1941.

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Stieltjes Integral

Cite this as:

Weisstein, Eric W. "Stieltjes Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StieltjesIntegral.html

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