# Difference between revisions of "Lebesgue-Stieltjes integral"

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A generalization of the [[Lebesgue integral|Lebesgue integral]]. For a non-negative measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579301.png" /> the name "Lebesgue–Stieltjes integral" is used in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579303.png" /> is not the Lebesgue measure; then the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579304.png" /> is defined in the same way as the Lebesgue integral in the general case. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579305.png" /> is of variable sign, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579306.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579308.png" /> are non-negative measures, and the Lebesgue–Stieltjes integral | A generalization of the [[Lebesgue integral|Lebesgue integral]]. For a non-negative measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579301.png" /> the name "Lebesgue–Stieltjes integral" is used in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579303.png" /> is not the Lebesgue measure; then the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579304.png" /> is defined in the same way as the Lebesgue integral in the general case. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579305.png" /> is of variable sign, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579306.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579308.png" /> are non-negative measures, and the Lebesgue–Stieltjes integral | ||

## Revision as of 11:12, 25 February 2013

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

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

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) |

**How to Cite This Entry:**

Lebesgue-Stieltjes integral.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Lebesgue-Stieltjes_integral&oldid=29489