Lebesgue-Stieltjes integral

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)