# 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 | ||

− | + | $$\int\limits_Xfd\mu=\int\limits_Xfd\mu_1-\int\limits_Xfd\mu_2,$$ | |

− | + | 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 | |

− | + | $$\int\limits_a^bfd\Phi.$$ | |

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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 13:07, 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://encyclopediaofmath.org/index.php?title=Lebesgue-Stieltjes_integral&oldid=29489