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

and consider the Riemann sum

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

or sometimes simply

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

(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 Resource. https://mathworld.wolfram.com/StieltjesIntegral.html

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