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Difference between revisions of "Lebesgue-Stieltjes integral"

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A generalization of the [[Lebesgue integral|Lebesgue integral]]. For a non-negative measure $\mu$ the name  "Lebesgue–Stieltjes integral"  is used in the case when $X=\mathbf R^n$ and $\mu$ is not the Lebesgue measure; then the integral $\int_Xfd\mu$ is defined in the same way as the Lebesgue integral in the general case. If $\mu$ is of variable sign, then $\mu=\mu_1-\mu_2$, where $\mu_1$ and $\mu_2$ 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
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$$\int\limits_Xfd\mu=\int\limits_Xfd\mu_1-\int\limits_Xfd\mu_2,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l0579309.png" /></td> </tr></table>
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under the condition that both integrals on the right-hand side exist. For $X=\mathbf R^1$ the fact that $\mu$ is countably additive and bounded is equivalent to the fact that the measure is generated by some function $\Phi$ of bounded variation. In this case the Lebesgue–Stieltjes integral is written in the form
  
under the condition that both integrals on the right-hand side exist. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l05793010.png" /> the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l05793011.png" /> is countably additive and bounded is equivalent to the fact that the measure is generated by some function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l05793012.png" /> of bounded variation. In this case the Lebesgue–Stieltjes integral is written in the form
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$$\int\limits_a^bfd\Phi.$$
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057930/l05793013.png" /></td> </tr></table>
 
  
 
For a discrete measure the Lebesgue–Stieltjes integral is a series of numbers.
 
For a discrete measure the Lebesgue–Stieltjes integral is a series of numbers.

Latest revision as of 15:07, 26 August 2014

A generalization of the Lebesgue integral. For a non-negative measure $\mu$ the name "Lebesgue–Stieltjes integral" is used in the case when $X=\mathbf R^n$ and $\mu$ is not the Lebesgue measure; then the integral $\int_Xfd\mu$ is defined in the same way as the Lebesgue integral in the general case. If $\mu$ is of variable sign, then $\mu=\mu_1-\mu_2$, where $\mu_1$ and $\mu_2$ are non-negative measures, and the Lebesgue–Stieltjes integral

$$\int\limits_Xfd\mu=\int\limits_Xfd\mu_1-\int\limits_Xfd\mu_2,$$

under the condition that both integrals on the right-hand side exist. For $X=\mathbf R^1$ the fact that $\mu$ is countably additive and bounded is equivalent to the fact that the measure is generated by some function $\Phi$ of bounded variation. In this case the Lebesgue–Stieltjes integral is written in the form

$$\int\limits_a^bfd\Phi.$$

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
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article