Lebesgue-Stieltjes Integral

Let $\alpha(x)$ be a monotone increasing function and define an Interval $I=(x_1, x_2)$. Then define the Nonnegative function

$$U(I)=\alpha(x_2+0)-\alpha(x_1+0).$$

The Lebesgue Integral with respect to a Measure constructed using $U(I)$ is called the Lebesgue-Stieltjes integral, or sometimes the Lebesgue-Radon Integral.

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 326, 1980.