Difference between revisions of "Lebesgue-Stieltjes integral"
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Revision as of 18:53, 24 March 2012
A generalization of the Lebesgue integral. For a non-negative measure the name "Lebesgue–Stieltjes integral" is used in the case when
and
is not the Lebesgue measure; then the integral
is defined in the same way as the Lebesgue integral in the general case. If
is of variable sign, then
, where
and
are non-negative measures, and the Lebesgue–Stieltjes integral
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under the condition that both integrals on the right-hand side exist. For the fact that
is countably additive and bounded is equivalent to the fact that the measure is generated by some function
of bounded variation. In this case the Lebesgue–Stieltjes integral is written in the form
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For a discrete measure the Lebesgue–Stieltjes integral is a series of numbers.
References
[1] | E. Kamke, "Das Lebesgue–Stieltjes-Integral" , Teubner (1960) |
Comments
References
[a1] | E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) |
Lebesgue-Stieltjes integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue-Stieltjes_integral&oldid=14842